Theorem atbtwn 38913

Description: Property of a 3rd atom 𝑅 on a line 𝑃 ∨ 𝑄 intersecting element 𝑋 at 𝑃. (Contributed by NM, 30-Jul-2012.)

Hypotheses

Ref	Expression
atbtwn.b	⊢ 𝐵 = (Base‘𝐾)
atbtwn.l	⊢ ≤ = (le‘𝐾)
atbtwn.j	⊢ ∨ = (join‘𝐾)
atbtwn.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atbtwn	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≠ 𝑃 ↔ ¬ 𝑅 ≤ 𝑋))

Proof of Theorem atbtwn

Step	Hyp	Ref	Expression
1		simpl33 1254	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑅 ≤ (𝑃 ∨ 𝑄))
2		simpr 484	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑅 ≤ 𝑋)
3		simpl11 1246	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝐾 ∈ HL)
4	3	hllatd 38830	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝐾 ∈ Lat)
5		simpl2l 1224	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑅 ∈ 𝐴)
6		atbtwn.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
7		atbtwn.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	atbase 38755	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
9	5, 8	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑅 ∈ 𝐵)
10		simpl1 1189	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
11		atbtwn.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
12	6, 11, 7	hlatjcl 38833	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
13	10, 12	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → (𝑃 ∨ 𝑄) ∈ 𝐵)
14		simpl2r 1225	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑋 ∈ 𝐵)
15		atbtwn.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
16		eqid 2728	. . . . . . . . 9 ⊢ (meet‘𝐾) = (meet‘𝐾)
17	6, 15, 16	latlem12 18451	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ 𝑋) ↔ 𝑅 ≤ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋)))
18	4, 9, 13, 14, 17	syl13anc 1370	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ 𝑋) ↔ 𝑅 ≤ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋)))
19	1, 2, 18	mpbi2and 711	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑅 ≤ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋))
20		simpl12 1247	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑃 ∈ 𝐴)
21		simpl13 1248	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑄 ∈ 𝐴)
22		simpl31 1252	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑃 ≤ 𝑋)
23		simpl32 1253	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → ¬ 𝑄 ≤ 𝑋)
24	6, 15, 11, 16, 7	2atjm 38912	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) = 𝑃)
25	3, 20, 21, 14, 22, 23, 24	syl132anc 1386	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) = 𝑃)
26	19, 25	breqtrd 5168	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑅 ≤ 𝑃)
27		hlatl 38826	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28	3, 27	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝐾 ∈ AtLat)
29	15, 7	atcmp 38777	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑅 ≤ 𝑃 ↔ 𝑅 = 𝑃))
30	28, 5, 20, 29	syl3anc 1369	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → (𝑅 ≤ 𝑃 ↔ 𝑅 = 𝑃))
31	26, 30	mpbid 231	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑅 = 𝑃)
32	31	ex 412	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ 𝑋 → 𝑅 = 𝑃))
33	32	necon3ad 2949	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≠ 𝑃 → ¬ 𝑅 ≤ 𝑋))
34		simp31 1207	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ 𝑋)
35		nbrne2 5162	. . . . 5 ⊢ ((𝑃 ≤ 𝑋 ∧ ¬ 𝑅 ≤ 𝑋) → 𝑃 ≠ 𝑅)
36	35	necomd 2992	. . . 4 ⊢ ((𝑃 ≤ 𝑋 ∧ ¬ 𝑅 ≤ 𝑋) → 𝑅 ≠ 𝑃)
37	36	ex 412	. . 3 ⊢ (𝑃 ≤ 𝑋 → (¬ 𝑅 ≤ 𝑋 → 𝑅 ≠ 𝑃))
38	34, 37	syl 17	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (¬ 𝑅 ≤ 𝑋 → 𝑅 ≠ 𝑃))
39	33, 38	impbid 211	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≠ 𝑃 ↔ ¬ 𝑅 ≤ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2936   class class class wbr 5142  ‘cfv 6542  (class class class)co 7414  Basecbs 17173  lecple 17233  joincjn 18296  meetcmee 18297  Latclat 18416  Atomscatm 38729  AtLatcal 38730  HLchlt 38816

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 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  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 5143  df-opab 5205  df-mpt 5226  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-proset 18280  df-poset 18298  df-plt 18315  df-lub 18331  df-glb 18332  df-join 18333  df-meet 18334  df-p0 18410  df-lat 18417  df-clat 18484  df-oposet 38642  df-ol 38644  df-oml 38645  df-covers 38732  df-ats 38733  df-atl 38764  df-cvlat 38788  df-hlat 38817

This theorem is referenced by:  atbtwnexOLDN  38914  atbtwnex  38915