Step |
Hyp |
Ref |
Expression |
1 |
|
mvrf.s |
|- S = ( I mPwSer R ) |
2 |
|
mvrf.v |
|- V = ( I mVar R ) |
3 |
|
mvrf.b |
|- B = ( Base ` S ) |
4 |
|
mvrf.i |
|- ( ph -> I e. W ) |
5 |
|
mvrf.r |
|- ( ph -> R e. Ring ) |
6 |
|
mvrf1.z |
|- .0. = ( 0g ` R ) |
7 |
|
mvrf1.o |
|- .1. = ( 1r ` R ) |
8 |
|
mvrf1.n |
|- ( ph -> .1. =/= .0. ) |
9 |
1 2 3 4 5
|
mvrf |
|- ( ph -> V : I --> B ) |
10 |
8
|
adantr |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> .1. =/= .0. ) |
11 |
|
simp2r |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( V ` x ) = ( V ` y ) ) |
12 |
11
|
fveq1d |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( V ` x ) ` ( z e. I |-> if ( z = x , 1 , 0 ) ) ) = ( ( V ` y ) ` ( z e. I |-> if ( z = x , 1 , 0 ) ) ) ) |
13 |
|
eqid |
|- { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
14 |
4
|
3ad2ant1 |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> I e. W ) |
15 |
5
|
3ad2ant1 |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> R e. Ring ) |
16 |
|
simp2ll |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> x e. I ) |
17 |
2 13 6 7 14 15 16
|
mvrid |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( V ` x ) ` ( z e. I |-> if ( z = x , 1 , 0 ) ) ) = .1. ) |
18 |
|
simp2lr |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> y e. I ) |
19 |
|
1nn0 |
|- 1 e. NN0 |
20 |
13
|
snifpsrbag |
|- ( ( I e. W /\ 1 e. NN0 ) -> ( z e. I |-> if ( z = x , 1 , 0 ) ) e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } ) |
21 |
14 19 20
|
sylancl |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( z e. I |-> if ( z = x , 1 , 0 ) ) e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } ) |
22 |
2 13 6 7 14 15 18 21
|
mvrval2 |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( V ` y ) ` ( z e. I |-> if ( z = x , 1 , 0 ) ) ) = if ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) , .1. , .0. ) ) |
23 |
12 17 22
|
3eqtr3d |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> .1. = if ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) , .1. , .0. ) ) |
24 |
|
simp3 |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> -. x = y ) |
25 |
|
mpteqb |
|- ( A. z e. I if ( z = x , 1 , 0 ) e. NN0 -> ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) <-> A. z e. I if ( z = x , 1 , 0 ) = if ( z = y , 1 , 0 ) ) ) |
26 |
|
0nn0 |
|- 0 e. NN0 |
27 |
19 26
|
ifcli |
|- if ( z = x , 1 , 0 ) e. NN0 |
28 |
27
|
a1i |
|- ( z e. I -> if ( z = x , 1 , 0 ) e. NN0 ) |
29 |
25 28
|
mprg |
|- ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) <-> A. z e. I if ( z = x , 1 , 0 ) = if ( z = y , 1 , 0 ) ) |
30 |
|
iftrue |
|- ( z = x -> if ( z = x , 1 , 0 ) = 1 ) |
31 |
|
eqeq1 |
|- ( z = x -> ( z = y <-> x = y ) ) |
32 |
31
|
ifbid |
|- ( z = x -> if ( z = y , 1 , 0 ) = if ( x = y , 1 , 0 ) ) |
33 |
30 32
|
eqeq12d |
|- ( z = x -> ( if ( z = x , 1 , 0 ) = if ( z = y , 1 , 0 ) <-> 1 = if ( x = y , 1 , 0 ) ) ) |
34 |
33
|
rspcv |
|- ( x e. I -> ( A. z e. I if ( z = x , 1 , 0 ) = if ( z = y , 1 , 0 ) -> 1 = if ( x = y , 1 , 0 ) ) ) |
35 |
29 34
|
syl5bi |
|- ( x e. I -> ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) -> 1 = if ( x = y , 1 , 0 ) ) ) |
36 |
16 35
|
syl |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) -> 1 = if ( x = y , 1 , 0 ) ) ) |
37 |
|
ax-1ne0 |
|- 1 =/= 0 |
38 |
|
eqeq1 |
|- ( 1 = if ( x = y , 1 , 0 ) -> ( 1 = 0 <-> if ( x = y , 1 , 0 ) = 0 ) ) |
39 |
38
|
necon3abid |
|- ( 1 = if ( x = y , 1 , 0 ) -> ( 1 =/= 0 <-> -. if ( x = y , 1 , 0 ) = 0 ) ) |
40 |
37 39
|
mpbii |
|- ( 1 = if ( x = y , 1 , 0 ) -> -. if ( x = y , 1 , 0 ) = 0 ) |
41 |
|
iffalse |
|- ( -. x = y -> if ( x = y , 1 , 0 ) = 0 ) |
42 |
40 41
|
nsyl2 |
|- ( 1 = if ( x = y , 1 , 0 ) -> x = y ) |
43 |
36 42
|
syl6 |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) -> x = y ) ) |
44 |
24 43
|
mtod |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> -. ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) ) |
45 |
|
iffalse |
|- ( -. ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) -> if ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) , .1. , .0. ) = .0. ) |
46 |
44 45
|
syl |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> if ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) , .1. , .0. ) = .0. ) |
47 |
23 46
|
eqtrd |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> .1. = .0. ) |
48 |
47
|
3expia |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> ( -. x = y -> .1. = .0. ) ) |
49 |
48
|
necon1ad |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> ( .1. =/= .0. -> x = y ) ) |
50 |
10 49
|
mpd |
|- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> x = y ) |
51 |
50
|
expr |
|- ( ( ph /\ ( x e. I /\ y e. I ) ) -> ( ( V ` x ) = ( V ` y ) -> x = y ) ) |
52 |
51
|
ralrimivva |
|- ( ph -> A. x e. I A. y e. I ( ( V ` x ) = ( V ` y ) -> x = y ) ) |
53 |
|
dff13 |
|- ( V : I -1-1-> B <-> ( V : I --> B /\ A. x e. I A. y e. I ( ( V ` x ) = ( V ` y ) -> x = y ) ) ) |
54 |
9 52 53
|
sylanbrc |
|- ( ph -> V : I -1-1-> B ) |