Theorem atbtwnexOLDN 40307

Description: There exists a 3rd atom 𝑟 on a line 𝑃 ∨ 𝑄 intersecting element 𝑋 at 𝑃, such that 𝑟 is different from 𝑄 and not in 𝑋. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
atbtwnexOLDN	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))

Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   𝐾,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑋,𝑟

Allowed substitution hint:   ∨ (𝑟)

Proof of Theorem atbtwnexOLDN

Step	Hyp	Ref	Expression
1		simpr2 1214	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ 𝑋)
2		simpr3 1215	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ¬ 𝑄 ≤ 𝑋)
3		nbrne2 5129	. . . 4 ⊢ ((𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋) → 𝑃 ≠ 𝑄)
4	1, 2, 3	syl2anc 596	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≠ 𝑄)
5		atbtwn.l	. . . 4 ⊢ ≤ = (le‘𝐾)
6		atbtwn.j	. . . 4 ⊢ ∨ = (join‘𝐾)
7		atbtwn.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
8	5, 6, 7	hlsupr 40246	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))
9	4, 8	syldan 603	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))
10		simp32 1229	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑟 ≠ 𝑄)
11		simp31 1228	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑟 ≠ 𝑃)
12		simp1l 1216	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
13		simp2 1155	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑟 ∈ 𝐴)
14		simp1r1 1288	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑋 ∈ 𝐵)
15		simp1r2 1289	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ 𝑋)
16		simp1r3 1290	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑄 ≤ 𝑋)
17		simp33 1230	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑟 ≤ (𝑃 ∨ 𝑄))
18		atbtwn.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
19	18, 5, 6, 7	atbtwn 40306	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝑟 ≠ 𝑃 ↔ ¬ 𝑟 ≤ 𝑋))
20	12, 13, 14, 15, 16, 17, 19	syl123anc 1414	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝑟 ≠ 𝑃 ↔ ¬ 𝑟 ≤ 𝑋))
21	11, 20	mpbid 235	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑟 ≤ 𝑋)
22	10, 21, 17	3jca 1146	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))
23	22	3exp 1137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑟 ∈ 𝐴 → ((𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)) → (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))))
24	23	reximdvai 3175	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))))
25	9, 24	mpd 16	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  lecple 17353  joincjn 18403  Atomscatm 40123  HLchlt 40210

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-proset 18386  df-poset 18405  df-plt 18420  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-p0 18515  df-lat 18524  df-clat 18591  df-oposet 40036  df-ol 40038  df-oml 40039  df-covers 40126  df-ats 40127  df-atl 40158  df-cvlat 40182  df-hlat 40211

This theorem is used by: (None)