Theorem trlcnv 40166

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 2730	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
2		eqid 2730	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
3		trlcnv.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4	1, 2, 3	lhpexnle 40007	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ (Atoms‘𝐾) ¬ 𝑝(le‘𝐾)𝑊)
5	4	adantr 480	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ∃𝑝 ∈ (Atoms‘𝐾) ¬ 𝑝(le‘𝐾)𝑊)
6		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		trlcnv.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	6, 3, 7	ltrn1o 40125	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9	8	3adant3 1132	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
10		simp3l 1202	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
11	6, 2	atbase 39289	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
12	10, 11	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝑝 ∈ (Base‘𝐾))
13		f1ocnvfv1 7254	. . . . . . . 8 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (◡𝐹‘(𝐹‘𝑝)) = 𝑝)
14	9, 12, 13	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (◡𝐹‘(𝐹‘𝑝)) = 𝑝)
15	14	oveq2d 7406	. . . . . 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 1198	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
17	1, 2, 3, 7	ltrnat 40141	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝐹‘𝑝) ∈ (Atoms‘𝐾))
18	17	3adant3r 1182	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐹‘𝑝) ∈ (Atoms‘𝐾))
19		eqid 2730	. . . . . . . 8 ⊢ (join‘𝐾) = (join‘𝐾)
20	19, 2	hlatjcom 39368	. . . . . . 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 1373	. . . . . 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 2765	. . . . 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 7405	. . . 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 1136	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
25	3, 7	ltrncnv 40147	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
26	25	3adant3 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ◡𝐹 ∈ 𝑇)
27	1, 2, 3, 7	ltrnel 40140	. . . . 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 2730	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
29		trlcnv.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
30	1, 19, 28, 2, 3, 7, 29	trlval2 40164	. . . . 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 1373	. . . 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 40164	. . . 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 2775	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅‘◡𝐹) = (𝑅‘𝐹))
34	33	3expa 1118	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅‘◡𝐹) = (𝑅‘𝐹))
35	5, 34	rexlimddv 3141	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘◡𝐹) = (𝑅‘𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3054   class class class wbr 5110  ^◡ccnv 5640  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  meetcmee 18280  Atomscatm 39263  HLchlt 39350  LHypclh 39985  LTrncltrn 40102  trLctrl 40159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-p1 18392  df-lat 18398  df-clat 18465  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-lhyp 39989  df-laut 39990  df-ldil 40105  df-ltrn 40106  df-trl 40160

This theorem is referenced by:  trlcocnv  40721  trlcoat  40724  trlcocnvat  40725  trlcone  40729  cdlemg46  40736  tendoicl  40797  cdlemh1  40816  cdlemh2  40817  cdlemh  40818  cdlemk3  40834  cdlemk12  40851  cdlemk12u  40873  cdlemkfid1N  40922  cdlemkid1  40923  cdlemkid2  40925  cdlemk45  40948