Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
|- ( x = 1 -> ( 1 x. x ) = ( 1 x. 1 ) ) |
2 |
|
id |
|- ( x = 1 -> x = 1 ) |
3 |
1 2
|
eqeq12d |
|- ( x = 1 -> ( ( 1 x. x ) = x <-> ( 1 x. 1 ) = 1 ) ) |
4 |
|
oveq2 |
|- ( x = y -> ( 1 x. x ) = ( 1 x. y ) ) |
5 |
|
id |
|- ( x = y -> x = y ) |
6 |
4 5
|
eqeq12d |
|- ( x = y -> ( ( 1 x. x ) = x <-> ( 1 x. y ) = y ) ) |
7 |
|
oveq2 |
|- ( x = ( y + 1 ) -> ( 1 x. x ) = ( 1 x. ( y + 1 ) ) ) |
8 |
|
id |
|- ( x = ( y + 1 ) -> x = ( y + 1 ) ) |
9 |
7 8
|
eqeq12d |
|- ( x = ( y + 1 ) -> ( ( 1 x. x ) = x <-> ( 1 x. ( y + 1 ) ) = ( y + 1 ) ) ) |
10 |
|
oveq2 |
|- ( x = A -> ( 1 x. x ) = ( 1 x. A ) ) |
11 |
|
id |
|- ( x = A -> x = A ) |
12 |
10 11
|
eqeq12d |
|- ( x = A -> ( ( 1 x. x ) = x <-> ( 1 x. A ) = A ) ) |
13 |
|
1t1e1ALT |
|- ( 1 x. 1 ) = 1 |
14 |
|
1cnd |
|- ( ( y e. NN /\ ( 1 x. y ) = y ) -> 1 e. CC ) |
15 |
|
simpl |
|- ( ( y e. NN /\ ( 1 x. y ) = y ) -> y e. NN ) |
16 |
15
|
nncnd |
|- ( ( y e. NN /\ ( 1 x. y ) = y ) -> y e. CC ) |
17 |
14 16 14
|
adddid |
|- ( ( y e. NN /\ ( 1 x. y ) = y ) -> ( 1 x. ( y + 1 ) ) = ( ( 1 x. y ) + ( 1 x. 1 ) ) ) |
18 |
|
simpr |
|- ( ( y e. NN /\ ( 1 x. y ) = y ) -> ( 1 x. y ) = y ) |
19 |
13
|
a1i |
|- ( ( y e. NN /\ ( 1 x. y ) = y ) -> ( 1 x. 1 ) = 1 ) |
20 |
18 19
|
oveq12d |
|- ( ( y e. NN /\ ( 1 x. y ) = y ) -> ( ( 1 x. y ) + ( 1 x. 1 ) ) = ( y + 1 ) ) |
21 |
17 20
|
eqtrd |
|- ( ( y e. NN /\ ( 1 x. y ) = y ) -> ( 1 x. ( y + 1 ) ) = ( y + 1 ) ) |
22 |
21
|
ex |
|- ( y e. NN -> ( ( 1 x. y ) = y -> ( 1 x. ( y + 1 ) ) = ( y + 1 ) ) ) |
23 |
3 6 9 12 13 22
|
nnind |
|- ( A e. NN -> ( 1 x. A ) = A ) |
24 |
|
nnre |
|- ( A e. NN -> A e. RR ) |
25 |
|
ax-1rid |
|- ( A e. RR -> ( A x. 1 ) = A ) |
26 |
24 25
|
syl |
|- ( A e. NN -> ( A x. 1 ) = A ) |
27 |
23 26
|
eqtr4d |
|- ( A e. NN -> ( 1 x. A ) = ( A x. 1 ) ) |