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Theorem 4atex2-0cOLDN 38656
Description: Same as 4atex2 38653 except that 𝑆 and 𝑇 are zero. TODO: do we need this one or 4atex2-0aOLDN 38654 or 4atex2-0bOLDN 38655? (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 ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 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
StepHypRef Expression
1 simp21l 1290 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝐴)
2 simp21r 1291 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑃 𝑊)
3 simp23 1208 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑆 = (0.‘𝐾))
43oveq1d 7399 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆 𝑃) = ((0.‘𝐾) 𝑃))
5 simp32 1210 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑇 = (0.‘𝐾))
65oveq1d 7399 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑇 𝑃) = ((0.‘𝐾) 𝑃))
74, 6eqtr4d 2774 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆 𝑃) = (𝑇 𝑃))
8 breq1 5135 . . . . 5 (𝑧 = 𝑃 → (𝑧 𝑊𝑃 𝑊))
98notbid 317 . . . 4 (𝑧 = 𝑃 → (¬ 𝑧 𝑊 ↔ ¬ 𝑃 𝑊))
10 oveq2 7392 . . . . 5 (𝑧 = 𝑃 → (𝑆 𝑧) = (𝑆 𝑃))
11 oveq2 7392 . . . . 5 (𝑧 = 𝑃 → (𝑇 𝑧) = (𝑇 𝑃))
1210, 11eqeq12d 2747 . . . 4 (𝑧 = 𝑃 → ((𝑆 𝑧) = (𝑇 𝑧) ↔ (𝑆 𝑃) = (𝑇 𝑃)))
139, 12anbi12d 631 . . 3 (𝑧 = 𝑃 → ((¬ 𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)) ↔ (¬ 𝑃 𝑊 ∧ (𝑆 𝑃) = (𝑇 𝑃))))
1413rspcev 3602 . 2 ((𝑃𝐴 ∧ (¬ 𝑃 𝑊 ∧ (𝑆 𝑃) = (𝑇 𝑃))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
151, 2, 7, 14syl12anc 835 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄𝑇 = (0.‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939  wrex 3069   class class class wbr 5132  cfv 6523  (class class class)co 7384  lecple 17176  joincjn 18236  0.cp0 18348  Atomscatm 37838  HLchlt 37925  LHypclh 38560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3426  df-v 3468  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-br 5133  df-iota 6475  df-fv 6531  df-ov 7387
This theorem is referenced by: (None)
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