Theorem trlcnv 40189

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 2736	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
2		eqid 2736	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
3		trlcnv.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4	1, 2, 3	lhpexnle 40030	. . 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 2736	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		trlcnv.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	6, 3, 7	ltrn1o 40148	. . . . . . . . 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 39312	. . . . . . . . 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 7274	. . . . . . . 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 7426	. . . . . 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 40164	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (join‘𝐾) = (join‘𝐾)
20	19, 2	hlatjcom 39391	. . . . . . 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 2771	. . . . 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 7425	. . . 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 40170	. . . . . 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 40163	. . . . 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 2736	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
29		trlcnv.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
30	1, 19, 28, 2, 3, 7, 29	trlval2 40187	. . . . 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 40187	. . . 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 2781	. . 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 3148	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 3061   class class class wbr 5124  ^◡ccnv 5658  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  lecple 17283  joincjn 18328  meetcmee 18329  Atomscatm 39286  HLchlt 39373  LHypclh 40008  LTrncltrn 40125  trLctrl 40182

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-proset 18311  df-poset 18330  df-plt 18345  df-lub 18361  df-glb 18362  df-join 18363  df-meet 18364  df-p0 18440  df-p1 18441  df-lat 18447  df-clat 18514  df-oposet 39199  df-ol 39201  df-oml 39202  df-covers 39289  df-ats 39290  df-atl 39321  df-cvlat 39345  df-hlat 39374  df-lhyp 40012  df-laut 40013  df-ldil 40128  df-ltrn 40129  df-trl 40183

This theorem is referenced by:  trlcocnv  40744  trlcoat  40747  trlcocnvat  40748  trlcone  40752  cdlemg46  40759  tendoicl  40820  cdlemh1  40839  cdlemh2  40840  cdlemh  40841  cdlemk3  40857  cdlemk12  40874  cdlemk12u  40896  cdlemkfid1N  40945  cdlemkid1  40946  cdlemkid2  40948  cdlemk45  40971