Theorem trljat1 39503

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 39201? (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 2731	. . . 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 39500	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊))
9	8	oveq1d 7427	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑅‘𝐹) ∨ 𝑃) = (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃))
10		simp1l 1196	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
11	10	hllatd 38700	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat)
12		simp3l 1200	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
13		eqid 2731	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 4	atbase 38625	. . . 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 39501	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
17	16	3adant3 1131	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) ∈ (Base‘𝐾))
18	13, 2	latjcom 18410	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ 𝑃))
19	11, 15, 17, 18	syl3anc 1370	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ 𝑃))
20	13, 5, 6	ltrncl 39462	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
21	15, 20	syld3an3 1408	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
22	13, 2	latjcl 18402	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
23	11, 15, 21, 22	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
24		simp1r 1197	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
25	13, 5	lhpbase 39335	. . . . 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 18411	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
28	11, 15, 21, 27	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
29	13, 1, 2, 3, 4	atmod2i1 39198	. . . 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 1382	. . 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 2731	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
32	1, 2, 31, 4, 5	lhpjat1 39357	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = (1.‘𝐾))
33	32	3adant2 1130	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = (1.‘𝐾))
34	33	oveq2d 7428	. . 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 38697	. . . . 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 38563	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃)))
38	36, 23, 37	syl2anc 583	. . 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 2776	. 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 2781	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   class class class wbr 5148  ‘cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274  meetcmee 18275  1.cp1 18387  Latclat 18394  OLcol 38510  Atomscatm 38599  HLchlt 38686  LHypclh 39321  LTrncltrn 39438  trLctrl 39495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-p1 18389  df-lat 18395  df-clat 18462  df-oposet 38512  df-ol 38514  df-oml 38515  df-covers 38602  df-ats 38603  df-atl 38634  df-cvlat 38658  df-hlat 38687  df-psubsp 38840  df-pmap 38841  df-padd 39133  df-lhyp 39325  df-laut 39326  df-ldil 39441  df-ltrn 39442  df-trl 39496

This theorem is referenced by:  trljat3  39505  trlval4  39525  trlval5  39526  cdlemc5  39532  cdlemk1  40168  cdlemk8  40175  cdlemki  40178  cdlemksv2  40184  cdlemk7  40185  cdlemk12  40187  cdlemk15  40192  cdlemk7u  40207  cdlemk12u  40209  cdlemk21N  40210  cdlemk20  40211  cdlemk22  40230  cdlemm10N  40455