Step |
Hyp |
Ref |
Expression |
1 |
|
sn-1ne2 |
|- 1 =/= 2 |
2 |
|
elre0re |
|- ( A e. RR -> 0 e. RR ) |
3 |
|
id |
|- ( A e. RR -> A e. RR ) |
4 |
2 3
|
remulcld |
|- ( A e. RR -> ( 0 x. A ) e. RR ) |
5 |
|
ax-rrecex |
|- ( ( ( 0 x. A ) e. RR /\ ( 0 x. A ) =/= 0 ) -> E. x e. RR ( ( 0 x. A ) x. x ) = 1 ) |
6 |
4 5
|
sylan |
|- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> E. x e. RR ( ( 0 x. A ) x. x ) = 1 ) |
7 |
|
simprr |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( ( 0 x. A ) x. x ) = 1 ) |
8 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
9 |
8
|
oveq1i |
|- ( 2 x. 0 ) = ( ( 1 + 1 ) x. 0 ) |
10 |
|
re0m0e0 |
|- ( 0 -R 0 ) = 0 |
11 |
10
|
eqcomi |
|- 0 = ( 0 -R 0 ) |
12 |
11
|
oveq2i |
|- ( ( 1 + 1 ) x. 0 ) = ( ( 1 + 1 ) x. ( 0 -R 0 ) ) |
13 |
|
1re |
|- 1 e. RR |
14 |
13 13
|
readdcli |
|- ( 1 + 1 ) e. RR |
15 |
|
sn-00idlem1 |
|- ( ( 1 + 1 ) e. RR -> ( ( 1 + 1 ) x. ( 0 -R 0 ) ) = ( ( 1 + 1 ) -R ( 1 + 1 ) ) ) |
16 |
14 15
|
ax-mp |
|- ( ( 1 + 1 ) x. ( 0 -R 0 ) ) = ( ( 1 + 1 ) -R ( 1 + 1 ) ) |
17 |
|
repnpcan |
|- ( ( 1 e. RR /\ 1 e. RR /\ 1 e. RR ) -> ( ( 1 + 1 ) -R ( 1 + 1 ) ) = ( 1 -R 1 ) ) |
18 |
13 13 13 17
|
mp3an |
|- ( ( 1 + 1 ) -R ( 1 + 1 ) ) = ( 1 -R 1 ) |
19 |
|
re1m1e0m0 |
|- ( 1 -R 1 ) = ( 0 -R 0 ) |
20 |
18 19 10
|
3eqtri |
|- ( ( 1 + 1 ) -R ( 1 + 1 ) ) = 0 |
21 |
12 16 20
|
3eqtri |
|- ( ( 1 + 1 ) x. 0 ) = 0 |
22 |
9 21
|
eqtr2i |
|- 0 = ( 2 x. 0 ) |
23 |
22
|
oveq1i |
|- ( 0 x. A ) = ( ( 2 x. 0 ) x. A ) |
24 |
23
|
oveq1i |
|- ( ( 0 x. A ) x. x ) = ( ( ( 2 x. 0 ) x. A ) x. x ) |
25 |
24
|
a1i |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( ( 0 x. A ) x. x ) = ( ( ( 2 x. 0 ) x. A ) x. x ) ) |
26 |
|
2cnd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> 2 e. CC ) |
27 |
|
0cnd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> 0 e. CC ) |
28 |
|
simpll |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> A e. RR ) |
29 |
28
|
recnd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> A e. CC ) |
30 |
26 27 29
|
mulassd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( ( 2 x. 0 ) x. A ) = ( 2 x. ( 0 x. A ) ) ) |
31 |
30
|
oveq1d |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( ( ( 2 x. 0 ) x. A ) x. x ) = ( ( 2 x. ( 0 x. A ) ) x. x ) ) |
32 |
4
|
ad2antrr |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( 0 x. A ) e. RR ) |
33 |
32
|
recnd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( 0 x. A ) e. CC ) |
34 |
|
simprl |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> x e. RR ) |
35 |
34
|
recnd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> x e. CC ) |
36 |
26 33 35
|
mulassd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( ( 2 x. ( 0 x. A ) ) x. x ) = ( 2 x. ( ( 0 x. A ) x. x ) ) ) |
37 |
25 31 36
|
3eqtrd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( ( 0 x. A ) x. x ) = ( 2 x. ( ( 0 x. A ) x. x ) ) ) |
38 |
7
|
oveq2d |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( 2 x. ( ( 0 x. A ) x. x ) ) = ( 2 x. 1 ) ) |
39 |
|
2re |
|- 2 e. RR |
40 |
|
ax-1rid |
|- ( 2 e. RR -> ( 2 x. 1 ) = 2 ) |
41 |
39 40
|
mp1i |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( 2 x. 1 ) = 2 ) |
42 |
37 38 41
|
3eqtrd |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> ( ( 0 x. A ) x. x ) = 2 ) |
43 |
7 42
|
eqtr3d |
|- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ ( x e. RR /\ ( ( 0 x. A ) x. x ) = 1 ) ) -> 1 = 2 ) |
44 |
6 43
|
rexlimddv |
|- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> 1 = 2 ) |
45 |
44
|
ex |
|- ( A e. RR -> ( ( 0 x. A ) =/= 0 -> 1 = 2 ) ) |
46 |
45
|
necon1d |
|- ( A e. RR -> ( 1 =/= 2 -> ( 0 x. A ) = 0 ) ) |
47 |
1 46
|
mpi |
|- ( A e. RR -> ( 0 x. A ) = 0 ) |