Theorem cdlemkfid3N 33908

Description: TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
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 ) ) ) )

Assertion

Ref	Expression
cdlemkfid3N	|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( G ` P ) )

Distinct variable groups:   ./\ , g   .\/ , g   B, g   P, g   R, g   T, g   g, Z   g, b   g, G

Allowed substitution hints:   A( g, b)   B( b)   P( b)   R( b)   T( b)   F( g, b)   G( b)   H( g, b)   .\/ ( b)   K( g, b)   .<_ ( g, b)   ./\ ( b)   N( g, b)   W( g, b)   Y( g, b)   Z( b)

Proof of Theorem cdlemkfid3N

Step	Hyp	Ref	Expression
1		simp22 1029	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
2		cdlemk5.y	. . . 4 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
3	2	cdlemk41 33903	. . 3 |- ( G e. T -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
4	1, 3	syl 17	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
5		simp1 995	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F = N ) )
6		simp21l 1112	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
7		simp21r 1113	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) )
8		simp23l 1116	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b e. T )
9		simp31 1031	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` b ) =/= ( R ` F ) )
10		simp33 1033	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11		cdlemk5.b	. . . . . 6 |- B = ( Base ` K )
12		cdlemk5.l	. . . . . 6 |- .<_ = ( le ` K )
13		cdlemk5.j	. . . . . 6 |- .\/ = ( join ` K )
14		cdlemk5.m	. . . . . 6 |- ./\ = ( meet ` K )
15		cdlemk5.a	. . . . . 6 |- A = ( Atoms ` K )
16		cdlemk5.h	. . . . . 6 |- H = ( LHyp ` K )
17		cdlemk5.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
18		cdlemk5.r	. . . . . 6 |- R = ( ( trL ` K ) ` W )
19		cdlemk5.z	. . . . . 6 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
20	11, 12, 13, 14, 15, 16, 17, 18, 19	cdlemkfid2N 33906	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) )
21	5, 6, 7, 8, 9, 10, 20	syl132anc 1246	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) )
22	21	oveq1d 6247	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( Z .\/ ( R ` ( G o. `' b ) ) ) = ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) )
23	22	oveq2d 6248	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )
24		simp1l 1019	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
25		simp23r 1117	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b =/= ( _I |` B ) )
26		simp32 1032	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` b ) =/= ( R ` G ) )
27	26	necomd 2672	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` G ) =/= ( R ` b ) )
28	11, 12, 13, 14, 15, 16, 17, 18	cdlemkfid1N 33904	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( b e. T /\ b =/= ( _I |` B ) /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` b ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( G ` P ) )
29	24, 8, 25, 1, 27, 10, 28	syl132anc 1246	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( G ` P ) )
30	4, 23, 29	3eqtrd 2445	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( G ` P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 972   = wceq 1403   e. wcel 1840   =/= wne 2596   [_csb 3370   class class class wbr 4392   _I cid 4730   `'ccnv 4939   |` cres 4942   o. ccom 4944   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   meetcmee 15788   Atomscatm 32245   HLchlt 32332   LHypclh 32965   LTrncltrn 33082   trLctrl 33140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-riotaBAD 31941

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-undef 6957  df-map 7377  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-llines 32479  df-lplanes 32480  df-lvols 32481  df-lines 32482  df-psubsp 32484  df-pmap 32485  df-padd 32777  df-lhyp 32969  df-laut 32970  df-ldil 33085  df-ltrn 33086  df-trl 33141

This theorem is referenced by: (None)