Theorem lhpmcvr6N 39412

Description: Specialization of lhpmcvr2 39408. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

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
lhpmcvr6N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋))

Distinct variable groups:   𝐴,𝑝   𝐵,𝑝   𝐾,𝑝   ≤ ,𝑝   ∧ ,𝑝   𝑋,𝑝   𝑊,𝑝   𝐻,𝑝   𝑌,𝑝

Allowed substitution hint:   ∨ (𝑝)

Proof of Theorem lhpmcvr6N

Step	Hyp	Ref	Expression
1		lhpmcvr2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		lhpmcvr2.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		lhpmcvr2.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		lhpmcvr2.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		lhpmcvr2.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		lhpmcvr2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7	1, 2, 3, 4, 5, 6	lhpmcvr5N 39411	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
8		simp31 1206	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ¬ 𝑝 ≤ 𝑊)
9		simp32 1207	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ¬ 𝑝 ≤ 𝑌)
10		simp11l 1281	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
11	10	hllatd 38747	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
12	1, 5	atbase 38672	. . . . . . . 8 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
13	12	3ad2ant2 1131	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑝 ∈ 𝐵)
14		simp12l 1283	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵)
15		simp11r 1282	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐻)
16	1, 6	lhpbase 39382	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
17	15, 16	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐵)
18	1, 4	latmcl 18405	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
19	11, 14, 17, 18	syl3anc 1368	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
20	1, 2, 3	latlej1 18413	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → 𝑝 ≤ (𝑝 ∨ (𝑋 ∧ 𝑊)))
21	11, 13, 19, 20	syl3anc 1368	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑝 ≤ (𝑝 ∨ (𝑋 ∧ 𝑊)))
22		simp33 1208	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
23	21, 22	breqtrd 5167	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑝 ≤ 𝑋)
24	8, 9, 23	3jca 1125	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋))
25	24	3expia 1118	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ 𝑝 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋)))
26	25	reximdva 3162	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋)))
27	7, 26	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∃wrex 3064   class class class wbr 5141  ‘cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  meetcmee 18277  Latclat 18396  Atomscatm 38646  HLchlt 38733  LHypclh 39368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-lhyp 39372

This theorem is referenced by:  dihmeetlem20N  40710