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Theorem 4atex2-0cOLDN 37286
Description: Same as 4atex2 37283 except that 𝑆 and 𝑇 are zero. TODO: do we need this one or 4atex2-0aOLDN 37284 or 4atex2-0bOLDN 37285? (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 1287 . 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 1288 . 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 1205 . . . 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 7160 . . 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 1207 . . . 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 7160 . . 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 2862 . 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 5055 . . . . 5 (𝑧 = 𝑃 → (𝑧 𝑊𝑃 𝑊))
98notbid 321 . . . 4 (𝑧 = 𝑃 → (¬ 𝑧 𝑊 ↔ ¬ 𝑃 𝑊))
10 oveq2 7153 . . . . 5 (𝑧 = 𝑃 → (𝑆 𝑧) = (𝑆 𝑃))
11 oveq2 7153 . . . . 5 (𝑧 = 𝑃 → (𝑇 𝑧) = (𝑇 𝑃))
1210, 11eqeq12d 2840 . . . 4 (𝑧 = 𝑃 → ((𝑆 𝑧) = (𝑇 𝑧) ↔ (𝑆 𝑃) = (𝑇 𝑃)))
139, 12anbi12d 633 . . 3 (𝑧 = 𝑃 → ((¬ 𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)) ↔ (¬ 𝑃 𝑊 ∧ (𝑆 𝑃) = (𝑇 𝑃))))
1413rspcev 3609 . 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 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wrex 3134   class class class wbr 5052  cfv 6343  (class class class)co 7145  lecple 16568  joincjn 17550  0.cp0 17643  Atomscatm 36469  HLchlt 36556  LHypclh 37190
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-rex 3139  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7148
This theorem is referenced by: (None)
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