Theorem cdlemg10bALTN 41128

Description: TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 41096 and pick best, if moved to ltrn* area. (Contributed by NM, 4-May-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg8.l	⊢ ≤ = (le‘𝐾)
cdlemg8.j	⊢ ∨ = (join‘𝐾)
cdlemg8.m	⊢ ∧ = (meet‘𝐾)
cdlemg8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemg10bALTN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊))

Proof of Theorem cdlemg10bALTN

Step	Hyp	Ref	Expression
1		simp11 1210	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐾 ∈ HL)
2		simp12 1211	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
3	1, 2	jca 516	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4		3simpc 1156	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
5		simp13 1212	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
6		cdlemg8.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemg8.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemg8.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
9		cdlemg8.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
10		cdlemg8.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
11		cdlemg8.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
12		eqid 2739	. . . . 5 ⊢ ((𝑃 ∨ 𝑄) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊)
13	6, 7, 8, 9, 10, 11, 12	cdlemg2k 41093	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
14	3, 4, 5, 13	syl3anc 1379	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
15	14	oveq1d 7371	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ∧ 𝑊) = (((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∧ 𝑊))
16		simp2 1143	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
17	8, 10, 6, 7	ltrnel 40631	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
18	3, 5, 16, 17	syl3anc 1379	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
19		eqid 2739	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
20	8, 11, 19, 10, 6	lhpmat 40522	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾))
21	3, 18, 20	syl2anc 590	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾))
22	21	oveq1d 7371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((𝐹‘𝑃) ∧ 𝑊) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((0.‘𝐾) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
23		simp2l 1206	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
24	8, 10, 6, 7	ltrnat 40632	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
25	3, 5, 23, 24	syl3anc 1379	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) ∈ 𝐴)
26	1	hllatd 39856	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐾 ∈ Lat)
27		simp3l 1208	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑄 ∈ 𝐴)
28		eqid 2739	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
29	28, 9, 10	hlatjcl 39859	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
30	1, 23, 27, 29	syl3anc 1379	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
31	28, 6	lhpbase 40490	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
32	2, 31	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾))
33	28, 11	latmcl 18397	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
34	26, 30, 32, 33	syl3anc 1379	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
35	28, 8, 11	latmle2 18422	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
36	26, 30, 32, 35	syl3anc 1379	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
37	28, 8, 9, 11, 10	atmod4i2 40359	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊) → (((𝐹‘𝑃) ∧ 𝑊) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = (((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∧ 𝑊))
38	1, 25, 34, 32, 36, 37	syl131anc 1391	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((𝐹‘𝑃) ∧ 𝑊) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = (((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∧ 𝑊))
39		hlol 39853	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
40	1, 39	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐾 ∈ OL)
41	28, 9, 19	olj02 39718	. . . 4 ⊢ ((𝐾 ∈ OL ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
42	40, 34, 41	syl2anc 590	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((0.‘𝐾) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
43	22, 38, 42	3eqtr3d 2782	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
44	15, 43	eqtrd 2774	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  lecple 17218  joincjn 18268  meetcmee 18269  0.cp0 18378  Latclat 18388  OLcol 39666  Atomscatm 39755  HLchlt 39842  LHypclh 40476  LTrncltrn 40593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-riotaBAD 39445

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-undef 8213  df-map 8765  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18389  df-clat 18456  df-oposet 39668  df-ol 39670  df-oml 39671  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-llines 39990  df-lplanes 39991  df-lvols 39992  df-lines 39993  df-psubsp 39995  df-pmap 39996  df-padd 40288  df-lhyp 40480  df-laut 40481  df-ldil 40596  df-ltrn 40597  df-trl 40651

This theorem is referenced by: (None)