Theorem 4atex2-0cOLDN 40874

Description: Same as 4atex2 40871 except that 𝑆 and 𝑇 are zero. TODO: do we need this one or 4atex2-0aOLDN 40872 or 4atex2-0bOLDN 40873? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
4that.l	⊢ ≤ = (le‘𝐾)
4that.j	⊢ ∨ = (join‘𝐾)
4that.a	⊢ 𝐴 = (Atoms‘𝐾)
4that.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
4atex2-0cOLDN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))

Distinct variable groups:   𝑧,𝑟,𝐴   𝐻,𝑟   ∨ ,𝑟,𝑧   𝐾,𝑟,𝑧   ≤ ,𝑟,𝑧   𝑃,𝑟,𝑧   𝑄,𝑟,𝑧   𝑆,𝑟,𝑧   𝑊,𝑟,𝑧   𝑇,𝑟,𝑧   𝑧,𝐻

Proof of Theorem 4atex2-0cOLDN

Step	Hyp	Ref	Expression
1		simp21l 1309	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑃 ∈ 𝐴)
2		simp21r 1310	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ¬ 𝑃 ≤ 𝑊)
3		simp23 1227	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑆 = (0.‘𝐾))
4	3	oveq1d 7425	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑆 ∨ 𝑃) = ((0.‘𝐾) ∨ 𝑃))
5		simp32 1229	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑇 = (0.‘𝐾))
6	5	oveq1d 7425	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑇 ∨ 𝑃) = ((0.‘𝐾) ∨ 𝑃))
7	4, 6	eqtr4d 2801	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑆 ∨ 𝑃) = (𝑇 ∨ 𝑃))
8		breq1 5112	. . . . 5 ⊢ (𝑧 = 𝑃 → (𝑧 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊))
9	8	notbid 321	. . . 4 ⊢ (𝑧 = 𝑃 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊))
10		oveq2 7418	. . . . 5 ⊢ (𝑧 = 𝑃 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑃))
11		oveq2 7418	. . . . 5 ⊢ (𝑧 = 𝑃 → (𝑇 ∨ 𝑧) = (𝑇 ∨ 𝑃))
12	10, 11	eqeq12d 2779	. . . 4 ⊢ (𝑧 = 𝑃 → ((𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧) ↔ (𝑆 ∨ 𝑃) = (𝑇 ∨ 𝑃)))
13	9, 12	anbi12d 643	. . 3 ⊢ (𝑧 = 𝑃 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)) ↔ (¬ 𝑃 ≤ 𝑊 ∧ (𝑆 ∨ 𝑃) = (𝑇 ∨ 𝑃))))
14	13	rspcev 3581	. 2 ⊢ ((𝑃 ∈ 𝐴 ∧ (¬ 𝑃 ≤ 𝑊 ∧ (𝑆 ∨ 𝑃) = (𝑇 ∨ 𝑃))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
15	1, 2, 7, 14	syl12anc 849	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 = (0.‘𝐾) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  lecple 17312  joincjn 18362  0.cp0 18472  Atomscatm 40057  HLchlt 40144  LHypclh 40778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by: (None)