Theorem lhpmcvr3 36692

Description: Specialization of lhpmcvr2 36691. TODO: Use this to simplify many uses of (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋 to become 𝑃 ≤ 𝑋. (Contributed by NM, 6-Apr-2014.)

Hypotheses

Ref	Expression
lhpmcvr2.b	⊢ 𝐵 = (Base‘𝐾)
lhpmcvr2.l	⊢ ≤ = (le‘𝐾)
lhpmcvr2.j	⊢ ∨ = (join‘𝐾)
lhpmcvr2.m	⊢ ∧ = (meet‘𝐾)
lhpmcvr2.a	⊢ 𝐴 = (Atoms‘𝐾)
lhpmcvr2.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
lhpmcvr3	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋))

Proof of Theorem lhpmcvr3

Step	Hyp	Ref	Expression
1		simpl1l 1217	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ HL)
2		simpl3l 1221	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴)
3		simpl2l 1219	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵)
4		simpl1r 1218	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑊 ∈ 𝐻)
5		lhpmcvr2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		lhpmcvr2.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
7	5, 6	lhpbase 36665	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
8	4, 7	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑊 ∈ 𝐵)
9		simpr 485	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑃 ≤ 𝑋)
10		lhpmcvr2.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
11		lhpmcvr2.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
12		lhpmcvr2.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
13		lhpmcvr2.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
14	5, 10, 11, 12, 13	atmod3i1 36531	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = (𝑋 ∧ (𝑃 ∨ 𝑊)))
15	1, 2, 3, 8, 9, 14	syl131anc 1376	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = (𝑋 ∧ (𝑃 ∨ 𝑊)))
16		simpl1 1184	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		simpl3 1186	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
18		eqid 2795	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
19	10, 11, 18, 13, 6	lhpjat2 36688	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
20	16, 17, 19	syl2anc 584	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
21	20	oveq2d 7032	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ (𝑃 ∨ 𝑊)) = (𝑋 ∧ (1.‘𝐾)))
22		hlol 36028	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
23	1, 22	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ OL)
24	5, 12, 18	olm11 35894	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ (1.‘𝐾)) = 𝑋)
25	23, 3, 24	syl2anc 584	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ (1.‘𝐾)) = 𝑋)
26	15, 21, 25	3eqtrd 2835	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
27		simpl1l 1217	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝐾 ∈ HL)
28	27	hllatd 36031	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝐾 ∈ Lat)
29		simpl3l 1221	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ∈ 𝐴)
30	5, 13	atbase 35956	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
31	29, 30	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ∈ 𝐵)
32		simpl2l 1219	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑋 ∈ 𝐵)
33		simpl1r 1218	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑊 ∈ 𝐻)
34	33, 7	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑊 ∈ 𝐵)
35	5, 12	latmcl 17491	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
36	28, 32, 34, 35	syl3anc 1364	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → (𝑋 ∧ 𝑊) ∈ 𝐵)
37	5, 10, 11	latlej1 17499	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ (𝑋 ∧ 𝑊)))
38	28, 31, 36, 37	syl3anc 1364	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ≤ (𝑃 ∨ (𝑋 ∧ 𝑊)))
39		simpr 485	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
40	38, 39	breqtrd 4988	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ≤ 𝑋)
41	26, 40	impbida 797	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   class class class wbr 4962  ‘cfv 6225  (class class class)co 7016  Basecbs 16312  lecple 16401  joincjn 17383  meetcmee 17384  1.cp1 17477  Latclat 17484  OLcol 35841  Atomscatm 35930  HLchlt 36017  LHypclh 36651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-proset 17367  df-poset 17385  df-plt 17397  df-lub 17413  df-glb 17414  df-join 17415  df-meet 17416  df-p0 17478  df-p1 17479  df-lat 17485  df-clat 17547  df-oposet 35843  df-ol 35845  df-oml 35846  df-covers 35933  df-ats 35934  df-atl 35965  df-cvlat 35989  df-hlat 36018  df-psubsp 36170  df-pmap 36171  df-padd 36463  df-lhyp 36655

This theorem is referenced by:  dihvalcq2  37914