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Theorem cdlemg4a 34187
Description: TODO: FIX COMMENT If fg(p) = p, then tr f = tr g. (Contributed by NM, 23-Apr-2013.)
Hypotheses
Ref Expression
cdlemg4.l |- .<_ = ( le ` K )
cdlemg4.a |- A = ( Atoms ` K )
cdlemg4.h |- H = ( LHyp ` K )
cdlemg4.t |- T = ( ( LTrn ` K ) ` W )
cdlemg4.r |- R = ( ( trL ` K ) ` W )
Assertion
Ref Expression
cdlemg4a |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( R ` F ) = ( R ` G ) )
Proof of Theorem cdlemg4a
StepHypRef Expression
1 simp3 1011 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( F ` ( G ` P ) ) = P )
21oveq2d 6311 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) = ( ( G ` P ) ( join ` K ) P ) )
3 simp1l 1033 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> K e. HL )
4 simp1 1009 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( K e. HL /\ W e. H ) )
5 simp23 1044 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> G e. T )
6 simp21 1042 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( P e. A /\ -. P .<_ W ) )
7 cdlemg4.l . . . . . . . 8 |- .<_ = ( le ` K )
8 cdlemg4.a . . . . . . . 8 |- A = ( Atoms ` K )
9 cdlemg4.h . . . . . . . 8 |- H = ( LHyp ` K )
10 cdlemg4.t . . . . . . . 8 |- T = ( ( LTrn ` K ) ` W )
117, 8, 9, 10ltrnel 33716 . . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
1211simpld 461 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( G ` P ) e. A )
134, 5, 6, 12syl3anc 1269 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( G ` P ) e. A )
14 simp21l 1126 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> P e. A )
15 eqid 2453 . . . . . 6 |- ( join ` K ) = ( join ` K )
1615, 8hlatjcom 32945 . . . . 5 |- ( ( K e. HL /\ ( G ` P ) e. A /\ P e. A ) -> ( ( G ` P ) ( join ` K ) P ) = ( P ( join ` K ) ( G ` P ) ) )
173, 13, 14, 16syl3anc 1269 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( G ` P ) ( join ` K ) P ) = ( P ( join ` K ) ( G ` P ) ) )
182, 17eqtrd 2487 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) = ( P ( join ` K ) ( G ` P ) ) )
1918oveq1d 6310 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) ( meet ` K ) W ) = ( ( P ( join ` K ) ( G ` P ) ) ( meet ` K ) W ) )
20 simp22 1043 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> F e. T )
214, 5, 6, 11syl3anc 1269 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
22 eqid 2453 . . . 4 |- ( meet ` K ) = ( meet ` K )
23 cdlemg4.r . . . 4 |- R = ( ( trL ` K ) ` W )
247, 15, 22, 8, 9, 10, 23trlval2 33741 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) ) -> ( R ` F ) = ( ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) ( meet ` K ) W ) )
254, 20, 21, 24syl3anc 1269 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( R ` F ) = ( ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) ( meet ` K ) W ) )
267, 15, 22, 8, 9, 10, 23trlval2 33741 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P ( join ` K ) ( G ` P ) ) ( meet ` K ) W ) )
274, 5, 6, 26syl3anc 1269 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( R ` G ) = ( ( P ( join ` K ) ( G ` P ) ) ( meet ` K ) W ) )
2819, 25, 273eqtr4d 2497 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( R ` F ) = ( R ` G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   lecple 15209   joincjn 16201   meetcmee 16202   Atomscatm 32841   HLchlt 32928   LHypclh 33561   LTrncltrn 33678   trLctrl 33736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7479  df-preset 16185  df-poset 16203  df-plt 16216  df-lub 16232  df-glb 16233  df-join 16234  df-meet 16235  df-p0 16297  df-lat 16304  df-oposet 32754  df-ol 32756  df-oml 32757  df-covers 32844  df-ats 32845  df-atl 32876  df-cvlat 32900  df-hlat 32929  df-lhyp 33565  df-laut 33566  df-ldil 33681  df-ltrn 33682  df-trl 33737
This theorem is referenced by:  cdlemg4f  34194
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