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Theorem 4atex2-0aOLDN 34883
Description: Same as 4atex2 34882 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 1184 . 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 1185 . 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 1083 . . . . 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 34167 . . . . 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 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
7 4that.a . . . . . 6 𝐴 = (Atoms‘𝐾)
86, 7atbase 34095 . . . . 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 2621 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
126, 10, 11olj02 34032 . . . 4 ((𝐾 ∈ OL ∧ 𝑇 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑇) = 𝑇)
135, 9, 12syl2anc 692 . . 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 1094 . . . 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 6630 . . 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 34174 . . . 4 ((𝐾 ∈ HL ∧ 𝑇𝐴) → (𝑇 𝑇) = 𝑇)
173, 1, 16syl2anc 692 . . 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 2665 . 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 4626 . . . . 5 (𝑧 = 𝑇 → (𝑧 𝑊𝑇 𝑊))
2019notbid 308 . . . 4 (𝑧 = 𝑇 → (¬ 𝑧 𝑊 ↔ ¬ 𝑇 𝑊))
21 oveq2 6623 . . . . 5 (𝑧 = 𝑇 → (𝑆 𝑧) = (𝑆 𝑇))
22 oveq2 6623 . . . . 5 (𝑧 = 𝑇 → (𝑇 𝑧) = (𝑇 𝑇))
2321, 22eqeq12d 2636 . . . 4 (𝑧 = 𝑇 → ((𝑆 𝑧) = (𝑇 𝑧) ↔ (𝑆 𝑇) = (𝑇 𝑇)))
2420, 23anbi12d 746 . . 3 (𝑧 = 𝑇 → ((¬ 𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)) ↔ (¬ 𝑇 𝑊 ∧ (𝑆 𝑇) = (𝑇 𝑇))))
2524rspcev 3299 . 2 ((𝑇𝐴 ∧ (¬ 𝑇 𝑊 ∧ (𝑆 𝑇) = (𝑇 𝑇))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
261, 2, 18, 25syl12anc 1321 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2909   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  lecple 15888  joincjn 16884  0.cp0 16977  OLcol 33980  Atomscatm 34069  HLchlt 34156  LHypclh 34789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-preset 16868  df-poset 16886  df-lub 16914  df-glb 16915  df-join 16916  df-meet 16917  df-p0 16979  df-lat 16986  df-oposet 33982  df-ol 33984  df-oml 33985  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157
This theorem is referenced by:  4atex2-0bOLDN  34884
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