Well I had to scratch my head on this one and dust off more than a few cobwebs. Now my understanding is that if you put a pulley on the ceiling and pass a rope through it to a 100 lbs weight and you hold that weight you are pulling on the free end of the rope with 100 lbs. If you are pulling straight down, it means you are are putting 100 lbs and the block is pulling down at 100 lbs and the pulley is exerting 200 lbs on the ceiling.
If you are pulling parallel to the ceiling then there is a 100 lbs down force and a 100 lbs horizontal force. or if you prefer a 141.4 lbs force at 45 degrees downward from the ceiling.
If my understanding is not valid, then the rest of this answer is not valid.
So I dropped this problem into S-Frame and placed the sum of the component horizontal and vertical loads as nodal loads. Because I can't just put 'a' in as a length I opted to make 'a' 10.
Now provided my understanding of pulleys is correct, then this model confirms there is no vertical load at A (point 1)
This is with the understanding that the vertical and horizontal components of th e 45 degree line break into roughly 70.7107 each
Sum of forces at point 7 is:
Fy=100 down from vertical + 70.7107 down from diagonal
Fx=70 right from diagonal
Sum of forces at point 4 is:
Fy=100 up from vertical
Sum of forces at point 5 is:
Fy= 100 down from vertical + 70.7107 up from diagonal
Fx= 70.7107 left from diagonal