Theorem cdleme0fN 39747

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme0.l	⊢ ≤ = (le‘𝐾)
cdleme0.j	⊢ ∨ = (join‘𝐾)
cdleme0.m	⊢ ∧ = (meet‘𝐾)
cdleme0.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme0.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme0.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme0c.3	⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊)

Assertion

Ref	Expression
cdleme0fN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑉 ≠ 𝑃)

Proof of Theorem cdleme0fN

Step	Hyp	Ref	Expression
1		cdleme0c.3	. . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊)
2		simp1l 1194	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ HL)
3	2	hllatd 38892	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ Lat)
4		simp2l 1196	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
5		eqid 2725	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		cdleme0.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7	5, 6	atbase 38817	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
8	4, 7	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑃 ∈ (Base‘𝐾))
9		simp3r 1199	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ 𝐴)
10	5, 6	atbase 38817	. . . . . 6 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
11	9, 10	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ (Base‘𝐾))
12		cdleme0.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
13	5, 12	latjcl 18430	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
14	3, 8, 11, 13	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
15		simp1r 1195	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑊 ∈ 𝐻)
16		cdleme0.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
17	5, 16	lhpbase 39527	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
18	15, 17	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑊 ∈ (Base‘𝐾))
19		cdleme0.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
20		cdleme0.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
21	5, 19, 20	latmle2 18456	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑅) ∧ 𝑊) ≤ 𝑊)
22	3, 14, 18, 21	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑅) ∧ 𝑊) ≤ 𝑊)
23	1, 22	eqbrtrid 5178	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑉 ≤ 𝑊)
24		simp2r 1197	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑃 ≤ 𝑊)
25		nbrne2 5163	. 2 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊) → 𝑉 ≠ 𝑃)
26	23, 24, 25	syl2anc 582	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑉 ≠ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2930   class class class wbr 5143  ‘cfv 6543  (class class class)co 7416  Basecbs 17179  lecple 17239  joincjn 18302  meetcmee 18303  Latclat 18422  Atomscatm 38791  HLchlt 38878  LHypclh 39513

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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738

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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-lat 18423  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879  df-lhyp 39517

This theorem is referenced by: (None)