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Theorem cdlemg4a 34591
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 990 . . . . 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 6217 . . . 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 1012 . . . . 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 988 . . . . . 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 1023 . . . . . 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 1021 . . . . . 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 34122 . . . . . . 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 1219 . . . . 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 1105 . . . . 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 2454 . . . . . 6 |- ( join ` K ) = ( join ` K )
1615, 8hlatjcom 33351 . . . . 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 1219 . . . 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 2495 . . 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 6216 . 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 1022 . . 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 1219 . . 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 2454 . . . 4 |- ( meet ` K ) = ( meet ` K )
23 cdlemg4.r . . . 4 |- R = ( ( trL ` K ) ` W )
247, 15, 22, 8, 9, 10, 23trlval2 34146 . . 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 1219 . 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 34146 . . 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 1219 . 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 2505 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 965   = wceq 1370   e. wcel 1758   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   lecple 14365   joincjn 15234   meetcmee 15235   Atomscatm 33247   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142
This theorem is referenced by:  cdlemg4f  34598
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