Theorem lhpmcvr3 40517

Description: Specialization of lhpmcvr2 40516. 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 1231	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ HL)
2		simpl3l 1235	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴)
3		simpl2l 1233	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵)
4		simpl1r 1232	. . . . 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 40490	. . . . 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 40356	. . . 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 1391	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = (𝑋 ∧ (𝑃 ∨ 𝑊)))
16		simpl1 1198	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		simpl3 1200	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
18		eqid 2739	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
19	10, 11, 18, 13, 6	lhpjat2 40513	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
20	16, 17, 19	syl2anc 590	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
21	20	oveq2d 7372	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ (𝑃 ∨ 𝑊)) = (𝑋 ∧ (1.‘𝐾)))
22		hlol 39853	. . . . 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 39719	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ (1.‘𝐾)) = 𝑋)
25	23, 3, 24	syl2anc 590	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ (1.‘𝐾)) = 𝑋)
26	15, 21, 25	3eqtrd 2778	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
27		simpl1l 1231	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝐾 ∈ HL)
28	27	hllatd 39856	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝐾 ∈ Lat)
29		simpl3l 1235	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ∈ 𝐴)
30	5, 13	atbase 39781	. . . . 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 1233	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑋 ∈ 𝐵)
33		simpl1r 1232	. . . . . 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 18397	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
36	28, 32, 34, 35	syl3anc 1379	. . . 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 18405	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ (𝑋 ∧ 𝑊)))
38	28, 31, 36, 37	syl3anc 1379	. . 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 5098	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ≤ 𝑋)
41	26, 40	impbida 806	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 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  1.cp1 18379  Latclat 18388  OLcol 39666  Atomscatm 39755  HLchlt 39842  LHypclh 40476

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

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  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-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-psubsp 39995  df-pmap 39996  df-padd 40288  df-lhyp 40480

This theorem is referenced by:  dihvalcq2  41739