Theorem cdlemb3 40862

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 40261? Then replace cdlemb2 40297 with it. This is a more general version of cdlemb2 40297 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 1192	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl2 1193	. . . 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 40861	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊))
8	1, 2, 7	syl2anc 584	. . 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 1137	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑃 = 𝑄)
12	11	oveq2d 7374	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑃 ∨ 𝑃) = (𝑃 ∨ 𝑄))
13		simp11l 1285	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ HL)
14		simp12l 1287	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑃 ∈ 𝐴)
15	4, 5	hlatjidm 39625	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑃 ∨ 𝑃) = 𝑃)
16	13, 14, 15	syl2anc 584	. . . . . . . . . . . 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 5110	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ 𝑟 ≤ 𝑃))
19		hlatl 39616	. . . . . . . . . . . 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 1138	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
22	3, 5	atcmp 39567	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑟 ≤ 𝑃 ↔ 𝑟 = 𝑃))
23	20, 21, 14, 22	syl3anc 1373	. . . . . . . . . 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 630	. . . . . 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 1118	. . . 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 3158	. . 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 1192	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simpl2 1193	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34		simpl3 1194	. . 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 40297	. . 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 1377	. 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 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  lecple 17184  joincjn 18234  Atomscatm 39519  AtLatcal 39520  HLchlt 39606  LHypclh 40240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-p1 18347  df-lat 18355  df-clat 18422  df-oposet 39432  df-ol 39434  df-oml 39435  df-covers 39522  df-ats 39523  df-atl 39554  df-cvlat 39578  df-hlat 39607  df-lhyp 40244

This theorem is referenced by:  cdlemg6e  40878