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Theorem 4atex2-0aOLDN 37218
Description: Same as 4atex2 37217 except that 𝑆 is zero. (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
4atex2-0aOLDN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃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
Allowed substitution hint:   𝐻(𝑧)

Proof of Theorem 4atex2-0aOLDN
StepHypRef Expression
1 simp32l 1294 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑇𝐴)
2 simp32r 1295 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑇 𝑊)
3 simp1l 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
4 hlol 36501 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
53, 4syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ OL)
6 eqid 2824 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
7 4that.a . . . . . 6 𝐴 = (Atoms‘𝐾)
86, 7atbase 36429 . . . . 5 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
91, 8syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑇 ∈ (Base‘𝐾))
10 4that.j . . . . 5 = (join‘𝐾)
11 eqid 2824 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
126, 10, 11olj02 36366 . . . 4 ((𝐾 ∈ OL ∧ 𝑇 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑇) = 𝑇)
135, 9, 12syl2anc 586 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((0.‘𝐾) 𝑇) = 𝑇)
14 simp23 1204 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑆 = (0.‘𝐾))
1514oveq1d 7174 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆 𝑇) = ((0.‘𝐾) 𝑇))
1610, 7hlatjidm 36509 . . . 4 ((𝐾 ∈ HL ∧ 𝑇𝐴) → (𝑇 𝑇) = 𝑇)
173, 1, 16syl2anc 586 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑇 𝑇) = 𝑇)
1813, 15, 173eqtr4d 2869 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆 𝑇) = (𝑇 𝑇))
19 breq1 5072 . . . . 5 (𝑧 = 𝑇 → (𝑧 𝑊𝑇 𝑊))
2019notbid 320 . . . 4 (𝑧 = 𝑇 → (¬ 𝑧 𝑊 ↔ ¬ 𝑇 𝑊))
21 oveq2 7167 . . . . 5 (𝑧 = 𝑇 → (𝑆 𝑧) = (𝑆 𝑇))
22 oveq2 7167 . . . . 5 (𝑧 = 𝑇 → (𝑇 𝑧) = (𝑇 𝑇))
2321, 22eqeq12d 2840 . . . 4 (𝑧 = 𝑇 → ((𝑆 𝑧) = (𝑇 𝑧) ↔ (𝑆 𝑇) = (𝑇 𝑇)))
2420, 23anbi12d 632 . . 3 (𝑧 = 𝑇 → ((¬ 𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)) ↔ (¬ 𝑇 𝑊 ∧ (𝑆 𝑇) = (𝑇 𝑇))))
2524rspcev 3626 . 2 ((𝑇𝐴 ∧ (¬ 𝑇 𝑊 ∧ (𝑆 𝑇) = (𝑇 𝑇))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
261, 2, 18, 25syl12anc 834 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆 = (0.‘𝐾)) ∧ (𝑃𝑄 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ ∃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 398  w3a 1083   = wceq 1536  wcel 2113  wne 3019  wrex 3142   class class class wbr 5069  cfv 6358  (class class class)co 7159  Basecbs 16486  lecple 16575  joincjn 17557  0.cp0 17650  OLcol 36314  Atomscatm 36403  HLchlt 36490  LHypclh 37124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-proset 17541  df-poset 17559  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-lat 17659  df-oposet 36316  df-ol 36318  df-oml 36319  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491
This theorem is referenced by:  4atex2-0bOLDN  37219
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