Step |
Hyp |
Ref |
Expression |
1 |
|
pj1eu.a |
|- .+ = ( +g ` G ) |
2 |
|
pj1eu.s |
|- .(+) = ( LSSum ` G ) |
3 |
|
pj1eu.o |
|- .0. = ( 0g ` G ) |
4 |
|
pj1eu.z |
|- Z = ( Cntz ` G ) |
5 |
|
pj1eu.2 |
|- ( ph -> T e. ( SubGrp ` G ) ) |
6 |
|
pj1eu.3 |
|- ( ph -> U e. ( SubGrp ` G ) ) |
7 |
|
pj1eu.4 |
|- ( ph -> ( T i^i U ) = { .0. } ) |
8 |
|
pj1eu.5 |
|- ( ph -> T C_ ( Z ` U ) ) |
9 |
|
pj1f.p |
|- P = ( proj1 ` G ) |
10 |
|
pj1eq.5 |
|- ( ph -> X e. ( T .(+) U ) ) |
11 |
|
pj1eq.6 |
|- ( ph -> B e. T ) |
12 |
|
pj1eq.7 |
|- ( ph -> C e. U ) |
13 |
1 2 3 4 5 6 7 8 9
|
pj1id |
|- ( ( ph /\ X e. ( T .(+) U ) ) -> X = ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) ) |
14 |
10 13
|
mpdan |
|- ( ph -> X = ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) ) |
15 |
14
|
eqeq1d |
|- ( ph -> ( X = ( B .+ C ) <-> ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) = ( B .+ C ) ) ) |
16 |
1 2 3 4 5 6 7 8 9
|
pj1f |
|- ( ph -> ( T P U ) : ( T .(+) U ) --> T ) |
17 |
16 10
|
ffvelrnd |
|- ( ph -> ( ( T P U ) ` X ) e. T ) |
18 |
1 2 3 4 5 6 7 8 9
|
pj2f |
|- ( ph -> ( U P T ) : ( T .(+) U ) --> U ) |
19 |
18 10
|
ffvelrnd |
|- ( ph -> ( ( U P T ) ` X ) e. U ) |
20 |
1 3 4 5 6 7 8 17 11 19 12
|
subgdisjb |
|- ( ph -> ( ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) = ( B .+ C ) <-> ( ( ( T P U ) ` X ) = B /\ ( ( U P T ) ` X ) = C ) ) ) |
21 |
15 20
|
bitrd |
|- ( ph -> ( X = ( B .+ C ) <-> ( ( ( T P U ) ` X ) = B /\ ( ( U P T ) ` X ) = C ) ) ) |