Theorem trljat1 40275

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 39973? (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 40272	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊))
9	8	oveq1d 7361	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑅‘𝐹) ∨ 𝑃) = (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃))
10		simp1l 1198	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
11	10	hllatd 39473	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat)
12		simp3l 1202	. . . 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 39398	. . . 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 40273	. . . 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 18353	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ 𝑃))
19	11, 15, 17, 18	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ 𝑃))
20	13, 5, 6	ltrncl 40234	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
21	15, 20	syld3an3 1411	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
22	13, 2	latjcl 18345	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
23	11, 15, 21, 22	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
24		simp1r 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
25	13, 5	lhpbase 40107	. . . . 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 18354	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
28	11, 15, 21, 27	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
29	13, 1, 2, 3, 4	atmod2i1 39970	. . . 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 1385	. . 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 40129	. . . . 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 7362	. . 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 39470	. . . . 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 39336	. . . 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 2771	. 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 2776	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 1541   ∈ wcel 2111   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  lecple 17168  joincjn 18217  meetcmee 18218  1.cp1 18328  Latclat 18337  OLcol 39283  Atomscatm 39372  HLchlt 39459  LHypclh 40093  LTrncltrn 40210  trLctrl 40267

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-p1 18330  df-lat 18338  df-clat 18405  df-oposet 39285  df-ol 39287  df-oml 39288  df-covers 39375  df-ats 39376  df-atl 39407  df-cvlat 39431  df-hlat 39460  df-psubsp 39612  df-pmap 39613  df-padd 39905  df-lhyp 40097  df-laut 40098  df-ldil 40213  df-ltrn 40214  df-trl 40268

This theorem is referenced by:  trljat3  40277  trlval4  40297  trlval5  40298  cdlemc5  40304  cdlemk1  40940  cdlemk8  40947  cdlemki  40950  cdlemksv2  40956  cdlemk7  40957  cdlemk12  40959  cdlemk15  40964  cdlemk7u  40979  cdlemk12u  40981  cdlemk21N  40982  cdlemk20  40983  cdlemk22  41002  cdlemm10N  41227