Metamath Proof Explorer


Theorem cdlemk19u1

Description: cdlemk19 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 31-Jul-2013)

Ref Expression
Hypotheses cdlemk5.b
|- B = ( Base ` K )
cdlemk5.l
|- .<_ = ( le ` K )
cdlemk5.j
|- .\/ = ( join ` K )
cdlemk5.m
|- ./\ = ( meet ` K )
cdlemk5.a
|- A = ( Atoms ` K )
cdlemk5.h
|- H = ( LHyp ` K )
cdlemk5.t
|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r
|- R = ( ( trL ` K ) ` W )
cdlemk5.z
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y
|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk5.x
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
cdlemk5.u
|- U = ( g e. T |-> if ( F = N , g , X ) )
Assertion cdlemk19u1
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( U ` F ) ` P ) = ( N ` P ) )

Proof

Step Hyp Ref Expression
1 cdlemk5.b
 |-  B = ( Base ` K )
2 cdlemk5.l
 |-  .<_ = ( le ` K )
3 cdlemk5.j
 |-  .\/ = ( join ` K )
4 cdlemk5.m
 |-  ./\ = ( meet ` K )
5 cdlemk5.a
 |-  A = ( Atoms ` K )
6 cdlemk5.h
 |-  H = ( LHyp ` K )
7 cdlemk5.t
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk5.r
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk5.z
 |-  Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10 cdlemk5.y
 |-  Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
11 cdlemk5.x
 |-  X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
12 cdlemk5.u
 |-  U = ( g e. T |-> if ( F = N , g , X ) )
13 simp22
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> F =/= N )
14 simp21
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
15 11 12 cdlemk40f
 |-  ( ( F =/= N /\ F e. T ) -> ( U ` F ) = [_ F / g ]_ X )
16 13 14 15 syl2anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( U ` F ) = [_ F / g ]_ X )
17 16 fveq1d
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( U ` F ) ` P ) = ( [_ F / g ]_ X ` P ) )
18 simp1l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
19 simp23
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> N e. T )
20 simp1r
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( R ` N ) )
21 1 6 7 8 trlnid
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( F =/= N /\ ( R ` F ) = ( R ` N ) ) ) -> F =/= ( _I |` B ) )
22 18 14 19 13 20 21 syl122anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> F =/= ( _I |` B ) )
23 14 22 19 3jca
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) )
24 1 2 3 4 5 6 7 8 9 10 11 cdlemk19x
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) )
25 23 24 syld3an2
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) )
26 17 25 eqtrd
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( U ` F ) ` P ) = ( N ` P ) )