Step |
Hyp |
Ref |
Expression |
1 |
|
neicvg.o |
|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) ) |
2 |
|
neicvg.p |
|- P = ( n e. _V |-> ( p e. ( ~P n ^m ~P n ) |-> ( o e. ~P n |-> ( n \ ( p ` ( n \ o ) ) ) ) ) ) |
3 |
|
neicvg.d |
|- D = ( P ` B ) |
4 |
|
neicvg.f |
|- F = ( ~P B O B ) |
5 |
|
neicvg.g |
|- G = ( B O ~P B ) |
6 |
|
neicvg.h |
|- H = ( F o. ( D o. G ) ) |
7 |
|
neicvg.r |
|- ( ph -> N H M ) |
8 |
6
|
cnveqi |
|- `' H = `' ( F o. ( D o. G ) ) |
9 |
|
cnvco |
|- `' ( F o. ( D o. G ) ) = ( `' ( D o. G ) o. `' F ) |
10 |
|
cnvco |
|- `' ( D o. G ) = ( `' G o. `' D ) |
11 |
10
|
coeq1i |
|- ( `' ( D o. G ) o. `' F ) = ( ( `' G o. `' D ) o. `' F ) |
12 |
8 9 11
|
3eqtri |
|- `' H = ( ( `' G o. `' D ) o. `' F ) |
13 |
3 6 7
|
neicvgbex |
|- ( ph -> B e. _V ) |
14 |
13
|
pwexd |
|- ( ph -> ~P B e. _V ) |
15 |
1 13 14 5 4
|
fsovcnvd |
|- ( ph -> `' G = F ) |
16 |
2 3 13
|
dssmapnvod |
|- ( ph -> `' D = D ) |
17 |
15 16
|
coeq12d |
|- ( ph -> ( `' G o. `' D ) = ( F o. D ) ) |
18 |
1 14 13 4 5
|
fsovcnvd |
|- ( ph -> `' F = G ) |
19 |
17 18
|
coeq12d |
|- ( ph -> ( ( `' G o. `' D ) o. `' F ) = ( ( F o. D ) o. G ) ) |
20 |
12 19
|
syl5eq |
|- ( ph -> `' H = ( ( F o. D ) o. G ) ) |
21 |
|
coass |
|- ( ( F o. D ) o. G ) = ( F o. ( D o. G ) ) |
22 |
21 6
|
eqtr4i |
|- ( ( F o. D ) o. G ) = H |
23 |
20 22
|
eqtrdi |
|- ( ph -> `' H = H ) |