Theorem atbtwn 39716

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 1257	. . . . . . 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 1249	. . . . . . . . 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 39634	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝐾 ∈ Lat)
5		simpl2l 1227	. . . . . . . . 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 39559	. . . . . . . . 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 1192	. . . . . . . . 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 39637	. . . . . . . . 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 1228	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (meet‘𝐾) = (meet‘𝐾)
17	6, 15, 16	latlem12 18389	. . . . . . . 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 1374	. . . . . . 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 712	. . . . . 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 1250	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑃 ∈ 𝐴)
21		simpl13 1251	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑄 ∈ 𝐴)
22		simpl31 1255	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑃 ≤ 𝑋)
23		simpl32 1256	. . . . . . 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 39715	. . . . . . 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 1390	. . . . . 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 5124	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≤ 𝑋) → 𝑅 ≤ 𝑃)
27		hlatl 39630	. . . . . . 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 39581	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑅 ≤ 𝑃 ↔ 𝑅 = 𝑃))
30	28, 5, 20, 29	syl3anc 1373	. . . . 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 232	. . . 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 2945	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≠ 𝑃 → ¬ 𝑅 ≤ 𝑋))
34		simp31 1210	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ 𝑋)
35		nbrne2 5118	. . . . 5 ⊢ ((𝑃 ≤ 𝑋 ∧ ¬ 𝑅 ≤ 𝑋) → 𝑃 ≠ 𝑅)
36	35	necomd 2987	. . . 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 212	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  meetcmee 18235  Latclat 18354  Atomscatm 39533  AtLatcal 39534  HLchlt 39620

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-lat 18355  df-clat 18422  df-oposet 39446  df-ol 39448  df-oml 39449  df-covers 39536  df-ats 39537  df-atl 39568  df-cvlat 39592  df-hlat 39621

This theorem is referenced by:  atbtwnexOLDN  39717  atbtwnex  39718