Theorem trlcnv 41315

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 2760	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
2		eqid 2760	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
3		trlcnv.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4	1, 2, 3	lhpexnle 41156	. . 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 2760	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		trlcnv.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	6, 3, 7	ltrn1o 41274	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9	8	3adant3 1150	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(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 40439	. . . . . . . . 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 7284	. . . . . . . 8 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(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 7436	. . . . . 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 41290	. . . . . . . 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 2760	. . . . . . . 8 ⊢ (join‘𝐾) = (join‘𝐾)
20	19, 2	hlatjcom 40518	. . . . . . 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 1398	. . . . . 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 2795	. . . . 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 7435	. . . 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 41296	. . . . . 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 41289	. . . . 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 2760	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
29		trlcnv.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
30	1, 19, 28, 2, 3, 7, 29	trlval2 41313	. . . . 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 1398	. . . 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 41313	. . . 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 2805	. . 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 3169	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 1569   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  ^◡ccnv 5649  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  lecple 17440  joincjn 18490  meetcmee 18491  Atomscatm 40413  HLchlt 40500  LHypclh 41134  LTrncltrn 41251  trLctrl 41308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8852  df-proset 18473  df-poset 18492  df-plt 18507  df-lub 18523  df-glb 18524  df-join 18525  df-meet 18526  df-p0 18602  df-p1 18603  df-lat 18611  df-clat 18678  df-oposet 40326  df-ol 40328  df-oml 40329  df-covers 40416  df-ats 40417  df-atl 40448  df-cvlat 40472  df-hlat 40501  df-lhyp 41138  df-laut 41139  df-ldil 41254  df-ltrn 41255  df-trl 41309

This theorem is used by:  trlcocnv  41870  trlcoat  41873  trlcocnvat  41874  trlcone  41878  cdlemg46  41885  tendoicl  41946  cdlemh1  41965  cdlemh2  41966  cdlemh  41967  cdlemk3  41983  cdlemk12  42000  cdlemk12u  42022  cdlemkfid1N  42071  cdlemkid1  42072  cdlemkid2  42074  cdlemk45  42097