Theorem trlcnv 41414

Description: The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)

Hypotheses

Ref	Expression
trlcnv.h	⊢ 𝐻 = (LHyp‘𝐾)
trlcnv.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlcnv.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
trlcnv	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘◡𝐹) = (𝑅‘𝐹))

Proof of Theorem trlcnv

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
2		eqid 2763	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
3		trlcnv.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4	1, 2, 3	lhpexnle 41255	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ (Atoms‘𝐾) ¬ 𝑝(le‘𝐾)𝑊)
5	4	adantr 486	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ∃𝑝 ∈ (Atoms‘𝐾) ¬ 𝑝(le‘𝐾)𝑊)
6		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		trlcnv.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	6, 3, 7	ltrn1o 41373	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)⟶1-1onto(Base‘𝐾))
9	8	3adant3 1150	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐹:(Base‘𝐾)⟶1-1onto(Base‘𝐾))
10		simp3l 1220	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
11	6, 2	atbase 40538	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
12	10, 11	syl 18	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝑝 ∈ (Base‘𝐾))
13		f1ocnvfv1 7307	. . . . . . . 8 ⊢ ((𝐹:(Base‘𝐾)⟶1-1onto(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (◡𝐹‘(𝐹‘𝑝)) = 𝑝)
14	9, 12, 13	syl2anc 596	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (◡𝐹‘(𝐹‘𝑝)) = 𝑝)
15	14	oveq2d 7459	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹‘𝑝)(join‘𝐾)(◡𝐹‘(𝐹‘𝑝))) = ((𝐹‘𝑝)(join‘𝐾)𝑝))
16		simp1l 1216	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
17	1, 2, 3, 7	ltrnat 41389	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝐹‘𝑝) ∈ (Atoms‘𝐾))
18	17	3adant3r 1200	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐹‘𝑝) ∈ (Atoms‘𝐾))
19		eqid 2763	. . . . . . . 8 ⊢ (join‘𝐾) = (join‘𝐾)
20	19, 2	hlatjcom 40617	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑝) ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝐹‘𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(𝐹‘𝑝)))
21	16, 18, 10, 20	syl3anc 1399	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹‘𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(𝐹‘𝑝)))
22	15, 21	eqtrd 2798	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹‘𝑝)(join‘𝐾)(◡𝐹‘(𝐹‘𝑝))) = (𝑝(join‘𝐾)(𝐹‘𝑝)))
23	22	oveq1d 7458	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (((𝐹‘𝑝)(join‘𝐾)(◡𝐹‘(𝐹‘𝑝)))(meet‘𝐾)𝑊) = ((𝑝(join‘𝐾)(𝐹‘𝑝))(meet‘𝐾)𝑊))
24		simp1 1154	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
25	3, 7	ltrncnv 41395	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
26	25	3adant3 1150	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ◡𝐹 ∈ 𝑇)
27	1, 2, 3, 7	ltrnel 41388	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹‘𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹‘𝑝)(le‘𝐾)𝑊))
28		eqid 2763	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
29		trlcnv.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
30	1, 19, 28, 2, 3, 7, 29	trlval2 41412	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ◡𝐹 ∈ 𝑇 ∧ ((𝐹‘𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹‘𝑝)(le‘𝐾)𝑊)) → (𝑅‘◡𝐹) = (((𝐹‘𝑝)(join‘𝐾)(◡𝐹‘(𝐹‘𝑝)))(meet‘𝐾)𝑊))
31	24, 26, 27, 30	syl3anc 1399	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅‘◡𝐹) = (((𝐹‘𝑝)(join‘𝐾)(◡𝐹‘(𝐹‘𝑝)))(meet‘𝐾)𝑊))
32	1, 19, 28, 2, 3, 7, 29	trlval2 41412	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅‘𝐹) = ((𝑝(join‘𝐾)(𝐹‘𝑝))(meet‘𝐾)𝑊))
33	23, 31, 32	3eqtr4d 2808	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅‘◡𝐹) = (𝑅‘𝐹))
34	33	3expa 1136	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅‘◡𝐹) = (𝑅‘𝐹))
35	5, 34	rexlimddv 3173	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘◡𝐹) = (𝑅‘𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5106  ^◡ccnv 5654  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  lecple 17464  joincjn 18514  meetcmee 18515  Atomscatm 40512  HLchlt 40599  LHypclh 41233  LTrncltrn 41350  trLctrl 41407

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-map 8877  df-proset 18497  df-poset 18516  df-plt 18531  df-lub 18547  df-glb 18548  df-join 18549  df-meet 18550  df-p0 18626  df-p1 18627  df-lat 18635  df-clat 18702  df-oposet 40425  df-ol 40427  df-oml 40428  df-covers 40515  df-ats 40516  df-atl 40547  df-cvlat 40571  df-hlat 40600  df-lhyp 41237  df-laut 41238  df-ldil 41353  df-ltrn 41354  df-trl 41408

This theorem is used by:  trlcocnv  41969  trlcoat  41972  trlcocnvat  41973  trlcone  41977  cdlemg46  41984  tendoicl  42045  cdlemh1  42064  cdlemh2  42065  cdlemh  42066  cdlemk3  42082  cdlemk12  42099  cdlemk12u  42121  cdlemkfid1N  42170  cdlemkid1  42171  cdlemkid2  42173  cdlemk45  42196