Step |
Hyp |
Ref |
Expression |
1 |
|
dpjfval.1 |
|- ( ph -> G dom DProd S ) |
2 |
|
dpjfval.2 |
|- ( ph -> dom S = I ) |
3 |
|
dpjfval.p |
|- P = ( G dProj S ) |
4 |
|
dpjfval.q |
|- Q = ( proj1 ` G ) |
5 |
|
df-dpj |
|- dProj = ( g e. Grp , s e. ( dom DProd " { g } ) |-> ( i e. dom s |-> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) ) ) |
6 |
5
|
a1i |
|- ( ph -> dProj = ( g e. Grp , s e. ( dom DProd " { g } ) |-> ( i e. dom s |-> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) ) ) ) |
7 |
|
simprr |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> s = S ) |
8 |
7
|
dmeqd |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> dom s = dom S ) |
9 |
2
|
adantr |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> dom S = I ) |
10 |
8 9
|
eqtrd |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> dom s = I ) |
11 |
|
simprl |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> g = G ) |
12 |
11
|
fveq2d |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> ( proj1 ` g ) = ( proj1 ` G ) ) |
13 |
12 4
|
eqtr4di |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> ( proj1 ` g ) = Q ) |
14 |
7
|
fveq1d |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> ( s ` i ) = ( S ` i ) ) |
15 |
10
|
difeq1d |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> ( dom s \ { i } ) = ( I \ { i } ) ) |
16 |
7 15
|
reseq12d |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> ( s |` ( dom s \ { i } ) ) = ( S |` ( I \ { i } ) ) ) |
17 |
11 16
|
oveq12d |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> ( g DProd ( s |` ( dom s \ { i } ) ) ) = ( G DProd ( S |` ( I \ { i } ) ) ) ) |
18 |
13 14 17
|
oveq123d |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) = ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) |
19 |
10 18
|
mpteq12dv |
|- ( ( ph /\ ( g = G /\ s = S ) ) -> ( i e. dom s |-> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) ) = ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) ) |
20 |
|
simpr |
|- ( ( ph /\ g = G ) -> g = G ) |
21 |
20
|
sneqd |
|- ( ( ph /\ g = G ) -> { g } = { G } ) |
22 |
21
|
imaeq2d |
|- ( ( ph /\ g = G ) -> ( dom DProd " { g } ) = ( dom DProd " { G } ) ) |
23 |
|
dprdgrp |
|- ( G dom DProd S -> G e. Grp ) |
24 |
1 23
|
syl |
|- ( ph -> G e. Grp ) |
25 |
|
reldmdprd |
|- Rel dom DProd |
26 |
|
elrelimasn |
|- ( Rel dom DProd -> ( S e. ( dom DProd " { G } ) <-> G dom DProd S ) ) |
27 |
25 26
|
ax-mp |
|- ( S e. ( dom DProd " { G } ) <-> G dom DProd S ) |
28 |
1 27
|
sylibr |
|- ( ph -> S e. ( dom DProd " { G } ) ) |
29 |
1 2
|
dprddomcld |
|- ( ph -> I e. _V ) |
30 |
29
|
mptexd |
|- ( ph -> ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) e. _V ) |
31 |
6 19 22 24 28 30
|
ovmpodx |
|- ( ph -> ( G dProj S ) = ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) ) |
32 |
3 31
|
syl5eq |
|- ( ph -> P = ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) ) |