Theorem lhp2atnle 37168

Description: Inequality for 2 different atoms under co-atom 𝑊. (Contributed by NM, 17-Jun-2013.)

Hypotheses

Ref	Expression
lhp2atnle.l	⊢ ≤ = (le‘𝐾)
lhp2atnle.j	⊢ ∨ = (join‘𝐾)
lhp2atnle.a	⊢ 𝐴 = (Atoms‘𝐾)
lhp2atnle.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
lhp2atnle	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ¬ 𝑉 ≤ (𝑃 ∨ 𝑈))

Proof of Theorem lhp2atnle

Step	Hyp	Ref	Expression
1		simp11l 1280	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝐾 ∈ HL)
2		hlatl 36495	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
3	1, 2	syl 17	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝐾 ∈ AtLat)
4		simp3l 1197	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ∈ 𝐴)
5		eqid 2821	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		lhp2atnle.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7	5, 6	atn0 36443	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑉 ∈ 𝐴) → 𝑉 ≠ (0.‘𝐾))
8	3, 4, 7	syl2anc 586	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ≠ (0.‘𝐾))
9	1	hllatd 36499	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝐾 ∈ Lat)
10		eqid 2821	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 6	atbase 36424	. . . . . 6 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
12	4, 11	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ∈ (Base‘𝐾))
13		simp12l 1282	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
14		simp2l 1195	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑈 ∈ 𝐴)
15		lhp2atnle.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
16	10, 15, 6	hlatjcl 36502	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
17	1, 13, 14, 16	syl3anc 1367	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
18		lhp2atnle.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
19		eqid 2821	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
20	10, 18, 19	latleeqm2 17689	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑉 ≤ (𝑃 ∨ 𝑈) ↔ ((𝑃 ∨ 𝑈)(meet‘𝐾)𝑉) = 𝑉))
21	9, 12, 17, 20	syl3anc 1367	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑉 ≤ (𝑃 ∨ 𝑈) ↔ ((𝑃 ∨ 𝑈)(meet‘𝐾)𝑉) = 𝑉))
22		lhp2atnle.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
23	18, 15, 19, 5, 6, 22	lhp2at0 37167	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑃 ∨ 𝑈)(meet‘𝐾)𝑉) = (0.‘𝐾))
24		eqeq1 2825	. . . . 5 ⊢ (((𝑃 ∨ 𝑈)(meet‘𝐾)𝑉) = 𝑉 → (((𝑃 ∨ 𝑈)(meet‘𝐾)𝑉) = (0.‘𝐾) ↔ 𝑉 = (0.‘𝐾)))
25	23, 24	syl5ibcom 247	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (((𝑃 ∨ 𝑈)(meet‘𝐾)𝑉) = 𝑉 → 𝑉 = (0.‘𝐾)))
26	21, 25	sylbid 242	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑉 = (0.‘𝐾)))
27	26	necon3ad 3029	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑉 ≠ (0.‘𝐾) → ¬ 𝑉 ≤ (𝑃 ∨ 𝑈)))
28	8, 27	mpd 15	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ¬ 𝑉 ≤ (𝑃 ∨ 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   class class class wbr 5065  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  lecple 16571  joincjn 17553  meetcmee 17554  0.cp0 17646  Latclat 17654  Atomscatm 36398  AtLatcal 36399  HLchlt 36485  LHypclh 37119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-proset 17537  df-poset 17555  df-plt 17567  df-lub 17583  df-glb 17584  df-join 17585  df-meet 17586  df-p0 17648  df-lat 17655  df-clat 17717  df-oposet 36311  df-ol 36313  df-oml 36314  df-covers 36401  df-ats 36402  df-atl 36433  df-cvlat 36457  df-hlat 36486  df-psubsp 36638  df-pmap 36639  df-padd 36931  df-lhyp 37123

This theorem is referenced by:  lhp2atne  37169  lhp2at0nle  37170  cdlemg27a  37827  cdlemg31c  37834  cdlemh  37952  cdlemk12  37985  cdlemk12u  38007