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Theorem cdlemg4a 35404
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 998 . . . . 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 6298 . . . 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 1020 . . . . 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 996 . . . . . 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 1031 . . . . . 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 1029 . . . . . 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 34935 . . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
1211simpld 459 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( G ` P ) e. A )
134, 5, 6, 12syl3anc 1228 . . . . 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 1113 . . . . 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 2467 . . . . . 6 |- ( join ` K ) = ( join ` K )
1615, 8hlatjcom 34164 . . . . 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 1228 . . . 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 2508 . . 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 6297 . 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 1030 . . 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 1228 . . 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 2467 . . . 4 |- ( meet ` K ) = ( meet ` K )
23 cdlemg4.r . . . 4 |- R = ( ( trL ` K ) ` W )
247, 15, 22, 8, 9, 10, 23trlval2 34959 . . 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 1228 . 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 34959 . . 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 1228 . 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 2518 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 369   /\ w3a 973   = wceq 1379   e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   lecple 14555   joincjn 15424   meetcmee 15425   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-lat 15526  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955
This theorem is referenced by:  cdlemg4f  35411
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