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Metamath Proof Explorer

Theorem cdlemk19x

Description: cdlemk19 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-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 ) )
Assertion 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 ) )

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 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
13 1 6 7 8 cdlemftr1 
 |-  ( ( K e. HL /\ W e. H ) -> E. b e. T ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) )
14 12 13 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> E. b e. T ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) )
15 nfv 
 |-  F/ b ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) )
16 nfcv 
 |-  F/_ b F
17 nfra1 
 |-  F/ b A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y )
18 nfcv 
 |-  F/_ b T
19 17 18 nfriota 
 |-  F/_ b ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
20 11 19 nfcxfr 
 |-  F/_ b X
21 16 20 nfcsbw 
 |-  F/_ b [_ F / g ]_ X
22 nfcv 
 |-  F/_ b P
23 21 22 nffv 
 |-  F/_ b ( [_ F / g ]_ X ` P )
24 23 nfeq1 
 |-  F/ b ( [_ F / g ]_ X ` P ) = ( N ` P )
25 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) )
26 simpl2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) )
27 simpl3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
28 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) )
29 1 2 3 4 5 6 7 8 9 10 11 cdlemk19xlem 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) )
30 25 26 27 28 29 syl121anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) )
31 30 exp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( b e. T -> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) ) ) )
32 15 24 31 rexlimd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( E. b e. T ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) ) )
33 14 32 mpd 
 |-  ( ( ( ( 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 ) )