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Theorem 4atex2-0cOLDN 39490
Description: Same as 4atex2 39487 except that 𝑆 and 𝑇 are zero. TODO: do we need this one or 4atex2-0aOLDN 39488 or 4atex2-0bOLDN 39489? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
4that.l ≀ = (leβ€˜πΎ)
4that.j ∨ = (joinβ€˜πΎ)
4that.a 𝐴 = (Atomsβ€˜πΎ)
4that.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
4atex2-0cOLDN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
Distinct variable groups:   𝑧,π‘Ÿ,𝐴   𝐻,π‘Ÿ   ∨ ,π‘Ÿ,𝑧   𝐾,π‘Ÿ,𝑧   ≀ ,π‘Ÿ,𝑧   𝑃,π‘Ÿ,𝑧   𝑄,π‘Ÿ,𝑧   𝑆,π‘Ÿ,𝑧   π‘Š,π‘Ÿ,𝑧   𝑇,π‘Ÿ,𝑧   𝑧,𝐻

Proof of Theorem 4atex2-0cOLDN
StepHypRef Expression
1 simp21l 1288 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑃 ∈ 𝐴)
2 simp21r 1289 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ Β¬ 𝑃 ≀ π‘Š)
3 simp23 1206 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑆 = (0.β€˜πΎ))
43oveq1d 7429 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝑆 ∨ 𝑃) = ((0.β€˜πΎ) ∨ 𝑃))
5 simp32 1208 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑇 = (0.β€˜πΎ))
65oveq1d 7429 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝑇 ∨ 𝑃) = ((0.β€˜πΎ) ∨ 𝑃))
74, 6eqtr4d 2770 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝑆 ∨ 𝑃) = (𝑇 ∨ 𝑃))
8 breq1 5145 . . . . 5 (𝑧 = 𝑃 β†’ (𝑧 ≀ π‘Š ↔ 𝑃 ≀ π‘Š))
98notbid 318 . . . 4 (𝑧 = 𝑃 β†’ (Β¬ 𝑧 ≀ π‘Š ↔ Β¬ 𝑃 ≀ π‘Š))
10 oveq2 7422 . . . . 5 (𝑧 = 𝑃 β†’ (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑃))
11 oveq2 7422 . . . . 5 (𝑧 = 𝑃 β†’ (𝑇 ∨ 𝑧) = (𝑇 ∨ 𝑃))
1210, 11eqeq12d 2743 . . . 4 (𝑧 = 𝑃 β†’ ((𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧) ↔ (𝑆 ∨ 𝑃) = (𝑇 ∨ 𝑃)))
139, 12anbi12d 630 . . 3 (𝑧 = 𝑃 β†’ ((Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)) ↔ (Β¬ 𝑃 ≀ π‘Š ∧ (𝑆 ∨ 𝑃) = (𝑇 ∨ 𝑃))))
1413rspcev 3607 . 2 ((𝑃 ∈ 𝐴 ∧ (Β¬ 𝑃 ≀ π‘Š ∧ (𝑆 ∨ 𝑃) = (𝑇 ∨ 𝑃))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
151, 2, 7, 14syl12anc 836 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 = (0.β€˜πΎ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆƒwrex 3065   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  lecple 17231  joincjn 18294  0.cp0 18406  Atomscatm 38672  HLchlt 38759  LHypclh 39394
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-ext 2698
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-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417
This theorem is referenced by: (None)
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