Theorem cdlemb3 40318

Description: Given two atoms not under the fiducial co-atom 𝑊, there is a third. Lemma B in [Crawley] p. 112. TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 39717? Then replace cdlemb2 39753 with it. This is a more general version of cdlemb2 39753 without 𝑃 ≠ 𝑄 condition. (Contributed by NM, 27-Apr-2013.)

Hypotheses

Ref	Expression
cdlemg5.l	⊢ ≤ = (le‘𝐾)
cdlemg5.j	⊢ ∨ = (join‘𝐾)
cdlemg5.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg5.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
cdlemb3	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))

Distinct variable groups:   𝐴,𝑟   𝐻,𝑟   𝐾,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑊,𝑟   ∨ ,𝑟   𝑄,𝑟

Proof of Theorem cdlemb3

Step	Hyp	Ref	Expression
1		simpl1 1188	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl2 1189	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		cdlemg5.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		cdlemg5.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		cdlemg5.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg5.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7	3, 4, 5, 6	cdlemg5 40317	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊))
8	1, 2, 7	syl2anc 582	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → ∃𝑟 ∈ 𝐴 (𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊))
9		ancom 459	. . . . . 6 ⊢ ((𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊) ↔ (¬ 𝑟 ≤ 𝑊 ∧ 𝑃 ≠ 𝑟))
10		eqcom 2733	. . . . . . . . 9 ⊢ (𝑃 = 𝑟 ↔ 𝑟 = 𝑃)
11		simp2 1134	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑃 = 𝑄)
12	11	oveq2d 7432	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 ∨ 𝑃) = (𝑃 ∨ 𝑄))
13		simp11l 1281	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ HL)
14		simp12l 1283	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑃 ∈ 𝐴)
15	4, 5	hlatjidm 39080	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑃 ∨ 𝑃) = 𝑃)
16	13, 14, 15	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 ∨ 𝑃) = 𝑃)
17	12, 16	eqtr3d 2768	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 ∨ 𝑄) = 𝑃)
18	17	breq2d 5157	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ 𝑟 ≤ 𝑃))
19		hlatl 39071	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
20	13, 19	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ AtLat)
21		simp3 1135	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
22	3, 5	atcmp 39022	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑟 ≤ 𝑃 ↔ 𝑟 = 𝑃))
23	20, 21, 14, 22	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑟 ≤ 𝑃 ↔ 𝑟 = 𝑃))
24	18, 23	bitr2d 279	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑟 = 𝑃 ↔ 𝑟 ≤ (𝑃 ∨ 𝑄)))
25	10, 24	bitrid 282	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 = 𝑟 ↔ 𝑟 ≤ (𝑃 ∨ 𝑄)))
26	25	necon3abid 2967	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 ≠ 𝑟 ↔ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
27	26	anbi2d 628	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → ((¬ 𝑟 ≤ 𝑊 ∧ 𝑃 ≠ 𝑟) ↔ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))))
28	9, 27	bitrid 282	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → ((𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊) ↔ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))))
29	28	3expa 1115	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) ∧ 𝑟 ∈ 𝐴) → ((𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊) ↔ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))))
30	29	rexbidva 3167	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → (∃𝑟 ∈ 𝐴 (𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊) ↔ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))))
31	8, 30	mpbid 231	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
32		simpl1 1188	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simpl2 1189	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34		simpl3 1190	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
35		simpr 483	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
36	3, 4, 5, 6	cdlemb2 39753	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
37	32, 33, 34, 35, 36	syl121anc 1372	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
38	31, 37	pm2.61dane 3019	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∃wrex 3060   class class class wbr 5145  ‘cfv 6546  (class class class)co 7416  lecple 17268  joincjn 18331  Atomscatm 38974  AtLatcal 38975  HLchlt 39061  LHypclh 39696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-proset 18315  df-poset 18333  df-plt 18350  df-lub 18366  df-glb 18367  df-join 18368  df-meet 18369  df-p0 18445  df-p1 18446  df-lat 18452  df-clat 18519  df-oposet 38887  df-ol 38889  df-oml 38890  df-covers 38977  df-ats 38978  df-atl 39009  df-cvlat 39033  df-hlat 39062  df-lhyp 39700

This theorem is referenced by:  cdlemg6e  40334