Theorem cdlemb3 41052

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 40451? Then replace cdlemb2 40487 with it. This is a more general version of cdlemb2 40487 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 1193	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl2 1194	. . . 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 41051	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊))
8	1, 2, 7	syl2anc 585	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → ∃𝑟 ∈ 𝐴 (𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊))
9		ancom 460	. . . . . 6 ⊢ ((𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊) ↔ (¬ 𝑟 ≤ 𝑊 ∧ 𝑃 ≠ 𝑟))
10		eqcom 2743	. . . . . . . . 9 ⊢ (𝑃 = 𝑟 ↔ 𝑟 = 𝑃)
11		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑃 = 𝑄)
12	11	oveq2d 7383	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 ∨ 𝑃) = (𝑃 ∨ 𝑄))
13		simp11l 1286	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ HL)
14		simp12l 1288	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑃 ∈ 𝐴)
15	4, 5	hlatjidm 39815	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑃 ∨ 𝑃) = 𝑃)
16	13, 14, 15	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 ∨ 𝑃) = 𝑃)
17	12, 16	eqtr3d 2773	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 ∨ 𝑄) = 𝑃)
18	17	breq2d 5097	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ 𝑟 ≤ 𝑃))
19		hlatl 39806	. . . . . . . . . . . 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 1139	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
22	3, 5	atcmp 39757	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑟 ≤ 𝑃 ↔ 𝑟 = 𝑃))
23	20, 21, 14, 22	syl3anc 1374	. . . . . . . . . 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 280	. . . . . . . . 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 283	. . . . . . . 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 2968	. . . . . . 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 631	. . . . . 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 283	. . . . 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 1119	. . . 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 3159	. . 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 232	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
32		simpl1 1193	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simpl2 1194	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34		simpl3 1195	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
35		simpr 484	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
36	3, 4, 5, 6	cdlemb2 40487	. . 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 1378	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  lecple 17227  joincjn 18277  Atomscatm 39709  AtLatcal 39710  HLchlt 39796  LHypclh 40430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-p1 18390  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797  df-lhyp 40434

This theorem is referenced by:  cdlemg6e  41068