Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
|- ( ( A x. 0 ) = 1 -> ( 2 x. ( A x. 0 ) ) = ( 2 x. 1 ) ) |
2 |
1
|
adantl |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> ( 2 x. ( A x. 0 ) ) = ( 2 x. 1 ) ) |
3 |
|
2re |
|- 2 e. RR |
4 |
|
ax-1rid |
|- ( 2 e. RR -> ( 2 x. 1 ) = 2 ) |
5 |
3 4
|
mp1i |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> ( 2 x. 1 ) = 2 ) |
6 |
2 5
|
eqtrd |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> ( 2 x. ( A x. 0 ) ) = 2 ) |
7 |
3
|
a1i |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> 2 e. RR ) |
8 |
|
simpl |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> A e. RR ) |
9 |
|
0red |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> 0 e. RR ) |
10 |
8 9
|
remulcld |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> ( A x. 0 ) e. RR ) |
11 |
7 10
|
remulcld |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> ( 2 x. ( A x. 0 ) ) e. RR ) |
12 |
|
sn-0ne2 |
|- 0 =/= 2 |
13 |
12
|
necomi |
|- 2 =/= 0 |
14 |
13
|
a1i |
|- ( ( 2 x. ( A x. 0 ) ) = 2 -> 2 =/= 0 ) |
15 |
|
eqtr2 |
|- ( ( ( 2 x. ( A x. 0 ) ) = 2 /\ ( 2 x. ( A x. 0 ) ) = 0 ) -> 2 = 0 ) |
16 |
14 15
|
mteqand |
|- ( ( 2 x. ( A x. 0 ) ) = 2 -> ( 2 x. ( A x. 0 ) ) =/= 0 ) |
17 |
|
ax-rrecex |
|- ( ( ( 2 x. ( A x. 0 ) ) e. RR /\ ( 2 x. ( A x. 0 ) ) =/= 0 ) -> E. x e. RR ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) |
18 |
11 16 17
|
syl2an |
|- ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) -> E. x e. RR ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) |
19 |
|
2cnd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> 2 e. CC ) |
20 |
|
simplll |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> A e. RR ) |
21 |
|
0red |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> 0 e. RR ) |
22 |
20 21
|
remulcld |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( A x. 0 ) e. RR ) |
23 |
22
|
recnd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( A x. 0 ) e. CC ) |
24 |
|
simprl |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> x e. RR ) |
25 |
24
|
recnd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> x e. CC ) |
26 |
19 23 25
|
mulassd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( ( 2 x. ( A x. 0 ) ) x. x ) = ( 2 x. ( ( A x. 0 ) x. x ) ) ) |
27 |
|
simprr |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) |
28 |
20
|
recnd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> A e. CC ) |
29 |
|
0cnd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> 0 e. CC ) |
30 |
28 29 25
|
mulassd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( ( A x. 0 ) x. x ) = ( A x. ( 0 x. x ) ) ) |
31 |
|
remul02 |
|- ( x e. RR -> ( 0 x. x ) = 0 ) |
32 |
31
|
ad2antrl |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( 0 x. x ) = 0 ) |
33 |
32
|
oveq2d |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( A x. ( 0 x. x ) ) = ( A x. 0 ) ) |
34 |
30 33
|
eqtrd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( ( A x. 0 ) x. x ) = ( A x. 0 ) ) |
35 |
34
|
oveq2d |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( 2 x. ( ( A x. 0 ) x. x ) ) = ( 2 x. ( A x. 0 ) ) ) |
36 |
26 27 35
|
3eqtr3rd |
|- ( ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) /\ ( x e. RR /\ ( ( 2 x. ( A x. 0 ) ) x. x ) = 1 ) ) -> ( 2 x. ( A x. 0 ) ) = 1 ) |
37 |
18 36
|
rexlimddv |
|- ( ( ( A e. RR /\ ( A x. 0 ) = 1 ) /\ ( 2 x. ( A x. 0 ) ) = 2 ) -> ( 2 x. ( A x. 0 ) ) = 1 ) |
38 |
6 37
|
mpdan |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> ( 2 x. ( A x. 0 ) ) = 1 ) |
39 |
|
sn-1ne2 |
|- 1 =/= 2 |
40 |
39
|
a1i |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> 1 =/= 2 ) |
41 |
38 40
|
eqnetrd |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> ( 2 x. ( A x. 0 ) ) =/= 2 ) |
42 |
6 41
|
pm2.21ddne |
|- ( ( A e. RR /\ ( A x. 0 ) = 1 ) -> -. ( A x. 0 ) = 1 ) |
43 |
42
|
ex |
|- ( A e. RR -> ( ( A x. 0 ) = 1 -> -. ( A x. 0 ) = 1 ) ) |
44 |
|
pm2.01 |
|- ( ( ( A x. 0 ) = 1 -> -. ( A x. 0 ) = 1 ) -> -. ( A x. 0 ) = 1 ) |
45 |
44
|
neqned |
|- ( ( ( A x. 0 ) = 1 -> -. ( A x. 0 ) = 1 ) -> ( A x. 0 ) =/= 1 ) |
46 |
43 45
|
syl |
|- ( A e. RR -> ( A x. 0 ) =/= 1 ) |
47 |
|
id |
|- ( A e. RR -> A e. RR ) |
48 |
|
elre0re |
|- ( A e. RR -> 0 e. RR ) |
49 |
47 48
|
remulcld |
|- ( A e. RR -> ( A x. 0 ) e. RR ) |
50 |
|
ax-rrecex |
|- ( ( ( A x. 0 ) e. RR /\ ( A x. 0 ) =/= 0 ) -> E. x e. RR ( ( A x. 0 ) x. x ) = 1 ) |
51 |
49 50
|
sylan |
|- ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) -> E. x e. RR ( ( A x. 0 ) x. x ) = 1 ) |
52 |
|
simpll |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> A e. RR ) |
53 |
52
|
recnd |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> A e. CC ) |
54 |
|
0cnd |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> 0 e. CC ) |
55 |
|
simprl |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> x e. RR ) |
56 |
55
|
recnd |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> x e. CC ) |
57 |
53 54 56
|
mulassd |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> ( ( A x. 0 ) x. x ) = ( A x. ( 0 x. x ) ) ) |
58 |
|
simprr |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> ( ( A x. 0 ) x. x ) = 1 ) |
59 |
31
|
oveq2d |
|- ( x e. RR -> ( A x. ( 0 x. x ) ) = ( A x. 0 ) ) |
60 |
59
|
ad2antrl |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> ( A x. ( 0 x. x ) ) = ( A x. 0 ) ) |
61 |
57 58 60
|
3eqtr3rd |
|- ( ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) /\ ( x e. RR /\ ( ( A x. 0 ) x. x ) = 1 ) ) -> ( A x. 0 ) = 1 ) |
62 |
51 61
|
rexlimddv |
|- ( ( A e. RR /\ ( A x. 0 ) =/= 0 ) -> ( A x. 0 ) = 1 ) |
63 |
62
|
ex |
|- ( A e. RR -> ( ( A x. 0 ) =/= 0 -> ( A x. 0 ) = 1 ) ) |
64 |
63
|
necon1d |
|- ( A e. RR -> ( ( A x. 0 ) =/= 1 -> ( A x. 0 ) = 0 ) ) |
65 |
46 64
|
mpd |
|- ( A e. RR -> ( A x. 0 ) = 0 ) |