Metamath Proof Explorer

Theorem cvmlift2lem8

Description: Lemma for cvmlift2 . (Contributed by Mario Carneiro, 9-Mar-2015)

Ref Expression
Hypotheses cvmlift2.b 
|- B = U. C
cvmlift2.f 
|- ( ph -> F e. ( C CovMap J ) )
cvmlift2.g 
|- ( ph -> G e. ( ( II tX II ) Cn J ) )
cvmlift2.p 
|- ( ph -> P e. B )
cvmlift2.i 
|- ( ph -> ( F ` P ) = ( 0 G 0 ) )
cvmlift2.h 
|- H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) )
cvmlift2.k 
|- K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) )
Assertion cvmlift2lem8 
|- ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( X K 0 ) = ( H ` X ) )

Proof

Step Hyp Ref Expression
1 cvmlift2.b 
 |-  B = U. C
2 cvmlift2.f 
 |-  ( ph -> F e. ( C CovMap J ) )
3 cvmlift2.g 
 |-  ( ph -> G e. ( ( II tX II ) Cn J ) )
4 cvmlift2.p 
 |-  ( ph -> P e. B )
5 cvmlift2.i 
 |-  ( ph -> ( F ` P ) = ( 0 G 0 ) )
6 cvmlift2.h 
 |-  H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) )
7 cvmlift2.k 
 |-  K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) )
8 simpr 
 |-  ( ( ph /\ X e. ( 0 [,] 1 ) ) -> X e. ( 0 [,] 1 ) )
9 0elunit 
 |-  0 e. ( 0 [,] 1 )
10 1 2 3 4 5 6 7 cvmlift2lem4 
 |-  ( ( X e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> ( X K 0 ) = ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` 0 ) )
11 8 9 10 sylancl 
 |-  ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( X K 0 ) = ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` 0 ) )
12 eqid 
 |-  ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) )
13 1 2 3 4 5 6 12 cvmlift2lem3 
 |-  ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) e. ( II Cn C ) /\ ( F o. ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` 0 ) = ( H ` X ) ) )
14 13 simp3d 
 |-  ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` 0 ) = ( H ` X ) )
15 11 14 eqtrd 
 |-  ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( X K 0 ) = ( H ` X ) )