Theorem lhpmcvr3 39538

Description: Specialization of lhpmcvr2 39537. 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 1221	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ HL)
2		simpl3l 1225	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴)
3		simpl2l 1223	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵)
4		simpl1r 1222	. . . . 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 39511	. . . . 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 483	. . . 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 39377	. . . 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 1380	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = (𝑋 ∧ (𝑃 ∨ 𝑊)))
16		simpl1 1188	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		simpl3 1190	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
18		eqid 2728	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
19	10, 11, 18, 13, 6	lhpjat2 39534	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
20	16, 17, 19	syl2anc 582	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
21	20	oveq2d 7442	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ (𝑃 ∨ 𝑊)) = (𝑋 ∧ (1.‘𝐾)))
22		hlol 38873	. . . . 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 38739	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ (1.‘𝐾)) = 𝑋)
25	23, 3, 24	syl2anc 582	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ (1.‘𝐾)) = 𝑋)
26	15, 21, 25	3eqtrd 2772	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
27		simpl1l 1221	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝐾 ∈ HL)
28	27	hllatd 38876	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝐾 ∈ Lat)
29		simpl3l 1225	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ∈ 𝐴)
30	5, 13	atbase 38801	. . . . 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 1223	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑋 ∈ 𝐵)
33		simpl1r 1222	. . . . . 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 18441	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
36	28, 32, 34, 35	syl3anc 1368	. . . 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 18449	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ (𝑋 ∧ 𝑊)))
38	28, 31, 36, 37	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ≤ (𝑃 ∨ (𝑋 ∧ 𝑊)))
39		simpr 483	. . 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 5178	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑃 ≤ 𝑋)
41	26, 40	impbida 799	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 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  ‘cfv 6553  (class class class)co 7426  Basecbs 17189  lecple 17249  joincjn 18312  meetcmee 18313  1.cp1 18425  Latclat 18432  OLcol 38686  Atomscatm 38775  HLchlt 38862  LHypclh 39497

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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-proset 18296  df-poset 18314  df-plt 18331  df-lub 18347  df-glb 18348  df-join 18349  df-meet 18350  df-p0 18426  df-p1 18427  df-lat 18433  df-clat 18500  df-oposet 38688  df-ol 38690  df-oml 38691  df-covers 38778  df-ats 38779  df-atl 38810  df-cvlat 38834  df-hlat 38863  df-psubsp 39016  df-pmap 39017  df-padd 39309  df-lhyp 39501

This theorem is referenced by:  dihvalcq2  40760