Theorem trljat1 39025

Description: The value of a translation of an atom 𝑃 not under the fiducial co-atom 𝑊, joined with trace. Equation above Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 38723? (Contributed by NM, 22-May-2012.)

Hypotheses

Ref	Expression
trljat.l	⊢ ≤ = (le‘𝐾)
trljat.j	⊢ ∨ = (join‘𝐾)
trljat.a	⊢ 𝐴 = (Atoms‘𝐾)
trljat.h	⊢ 𝐻 = (LHyp‘𝐾)
trljat.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trljat.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
trljat1	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))

Proof of Theorem trljat1

Step	Hyp	Ref	Expression
1		trljat.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		trljat.j	. . . 4 ⊢ ∨ = (join‘𝐾)
3		eqid 2732	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
4		trljat.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		trljat.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
6		trljat.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7		trljat.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	trlval2 39022	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊))
9	8	oveq1d 7420	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑅‘𝐹) ∨ 𝑃) = (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃))
10		simp1l 1197	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
11	10	hllatd 38222	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat)
12		simp3l 1201	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
13		eqid 2732	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 4	atbase 38147	. . . 4 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
15	12, 14	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ (Base‘𝐾))
16	13, 5, 6, 7	trlcl 39023	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
17	16	3adant3 1132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) ∈ (Base‘𝐾))
18	13, 2	latjcom 18396	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ 𝑃))
19	11, 15, 17, 18	syl3anc 1371	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ 𝑃))
20	13, 5, 6	ltrncl 38984	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
21	15, 20	syld3an3 1409	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
22	13, 2	latjcl 18388	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
23	11, 15, 21, 22	syl3anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
24		simp1r 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
25	13, 5	lhpbase 38857	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
26	24, 25	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾))
27	13, 1, 2	latlej1 18397	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
28	11, 15, 21, 27	syl3anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
29	13, 1, 2, 3, 4	atmod2i1 38720	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(𝑊 ∨ 𝑃)))
30	10, 12, 23, 26, 28, 29	syl131anc 1383	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(𝑊 ∨ 𝑃)))
31		eqid 2732	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
32	1, 2, 31, 4, 5	lhpjat1 38879	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = (1.‘𝐾))
33	32	3adant2 1131	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = (1.‘𝐾))
34	33	oveq2d 7421	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(𝑊 ∨ 𝑃)) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(1.‘𝐾)))
35		hlol 38219	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
36	10, 35	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ OL)
37	13, 3, 31	olm11 38085	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃)))
38	36, 23, 37	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃)))
39	30, 34, 38	3eqtrrd 2777	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) = (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃))
40	9, 19, 39	3eqtr4d 2782	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5147  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  meetcmee 18261  1.cp1 18373  Latclat 18380  OLcol 38032  Atomscatm 38121  HLchlt 38208  LHypclh 38843  LTrncltrn 38960  trLctrl 39017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-psubsp 38362  df-pmap 38363  df-padd 38655  df-lhyp 38847  df-laut 38848  df-ldil 38963  df-ltrn 38964  df-trl 39018

This theorem is referenced by:  trljat3  39027  trlval4  39047  trlval5  39048  cdlemc5  39054  cdlemk1  39690  cdlemk8  39697  cdlemki  39700  cdlemksv2  39706  cdlemk7  39707  cdlemk12  39709  cdlemk15  39714  cdlemk7u  39729  cdlemk12u  39731  cdlemk21N  39732  cdlemk20  39733  cdlemk22  39752  cdlemm10N  39977