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Theorem 4atex2-0aOLDN 38541
Description: Same as 4atex2 38540 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 1298 . 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 1299 . 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 1197 . . . . 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 37823 . . . . 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 2736 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
7 4that.a . . . . . 6 𝐴 = (Atoms‘𝐾)
86, 7atbase 37751 . . . . 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 2736 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
126, 10, 11olj02 37688 . . . 4 ((𝐾 ∈ OL ∧ 𝑇 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑇) = 𝑇)
135, 9, 12syl2anc 584 . . 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 1208 . . . 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 7372 . . 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 37831 . . . 4 ((𝐾 ∈ HL ∧ 𝑇𝐴) → (𝑇 𝑇) = 𝑇)
173, 1, 16syl2anc 584 . . 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 2786 . 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 5108 . . . . 5 (𝑧 = 𝑇 → (𝑧 𝑊𝑇 𝑊))
2019notbid 317 . . . 4 (𝑧 = 𝑇 → (¬ 𝑧 𝑊 ↔ ¬ 𝑇 𝑊))
21 oveq2 7365 . . . . 5 (𝑧 = 𝑇 → (𝑆 𝑧) = (𝑆 𝑇))
22 oveq2 7365 . . . . 5 (𝑧 = 𝑇 → (𝑇 𝑧) = (𝑇 𝑇))
2321, 22eqeq12d 2752 . . . 4 (𝑧 = 𝑇 → ((𝑆 𝑧) = (𝑇 𝑧) ↔ (𝑆 𝑇) = (𝑇 𝑇)))
2420, 23anbi12d 631 . . 3 (𝑧 = 𝑇 → ((¬ 𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)) ↔ (¬ 𝑇 𝑊 ∧ (𝑆 𝑇) = (𝑇 𝑇))))
2524rspcev 3581 . 2 ((𝑇𝐴 ∧ (¬ 𝑇 𝑊 ∧ (𝑆 𝑇) = (𝑇 𝑇))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑆 𝑧) = (𝑇 𝑧)))
261, 2, 18, 25syl12anc 835 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  0.cp0 18312  OLcol 37636  Atomscatm 37725  HLchlt 37812  LHypclh 38447
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-oposet 37638  df-ol 37640  df-oml 37641  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813
This theorem is referenced by:  4atex2-0bOLDN  38542
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