Theorem trlcnv 38106

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 2738	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
2		eqid 2738	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
3		trlcnv.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4	1, 2, 3	lhpexnle 37947	. . 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 2738	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		trlcnv.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	6, 3, 7	ltrn1o 38065	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9	8	3adant3 1130	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
10		simp3l 1199	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
11	6, 2	atbase 37230	. . . . . . . . 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 7129	. . . . . . . 8 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (◡𝐹‘(𝐹‘𝑝)) = 𝑝)
14	9, 12, 13	syl2anc 583	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (◡𝐹‘(𝐹‘𝑝)) = 𝑝)
15	14	oveq2d 7271	. . . . . 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 1195	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
17	1, 2, 3, 7	ltrnat 38081	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝐹‘𝑝) ∈ (Atoms‘𝐾))
18	17	3adant3r 1179	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐹‘𝑝) ∈ (Atoms‘𝐾))
19		eqid 2738	. . . . . . . 8 ⊢ (join‘𝐾) = (join‘𝐾)
20	19, 2	hlatjcom 37309	. . . . . . 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 1369	. . . . . 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 2778	. . . . 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 7270	. . . 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 1134	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
25	3, 7	ltrncnv 38087	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
26	25	3adant3 1130	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ◡𝐹 ∈ 𝑇)
27	1, 2, 3, 7	ltrnel 38080	. . . . 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 2738	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
29		trlcnv.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
30	1, 19, 28, 2, 3, 7, 29	trlval2 38104	. . . . 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 1369	. . . 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 38104	. . . 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 2788	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅‘◡𝐹) = (𝑅‘𝐹))
34	33	3expa 1116	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅‘◡𝐹) = (𝑅‘𝐹))
35	5, 34	rexlimddv 3219	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘◡𝐹) = (𝑅‘𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   class class class wbr 5070  ^◡ccnv 5579  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  joincjn 17944  meetcmee 17945  Atomscatm 37204  HLchlt 37291  LHypclh 37925  LTrncltrn 38042  trLctrl 38099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100

This theorem is referenced by:  trlcocnv  38661  trlcoat  38664  trlcocnvat  38665  trlcone  38669  cdlemg46  38676  tendoicl  38737  cdlemh1  38756  cdlemh2  38757  cdlemh  38758  cdlemk3  38774  cdlemk12  38791  cdlemk12u  38813  cdlemkfid1N  38862  cdlemkid1  38863  cdlemkid2  38865  cdlemk45  38888