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Theorem 4atex2-0aOLDN 39488
Description: Same as 4atex2 39487 except that 𝑆 is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
4that.l ≀ = (leβ€˜πΎ)
4that.j ∨ = (joinβ€˜πΎ)
4that.a 𝐴 = (Atomsβ€˜πΎ)
4that.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
4atex2-0aOLDN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
Distinct variable groups:   𝑧,π‘Ÿ,𝐴   𝐻,π‘Ÿ   ∨ ,π‘Ÿ,𝑧   𝐾,π‘Ÿ,𝑧   ≀ ,π‘Ÿ,𝑧   𝑃,π‘Ÿ,𝑧   𝑄,π‘Ÿ,𝑧   𝑆,π‘Ÿ,𝑧   π‘Š,π‘Ÿ,𝑧   𝑇,π‘Ÿ,𝑧
Allowed substitution hint:   𝐻(𝑧)

Proof of Theorem 4atex2-0aOLDN
StepHypRef Expression
1 simp32l 1296 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑇 ∈ 𝐴)
2 simp32r 1297 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ Β¬ 𝑇 ≀ π‘Š)
3 simp1l 1195 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ HL)
4 hlol 38770 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
53, 4syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ OL)
6 eqid 2727 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
7 4that.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
86, 7atbase 38698 . . . . 5 (𝑇 ∈ 𝐴 β†’ 𝑇 ∈ (Baseβ€˜πΎ))
91, 8syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑇 ∈ (Baseβ€˜πΎ))
10 4that.j . . . . 5 ∨ = (joinβ€˜πΎ)
11 eqid 2727 . . . . 5 (0.β€˜πΎ) = (0.β€˜πΎ)
126, 10, 11olj02 38635 . . . 4 ((𝐾 ∈ OL ∧ 𝑇 ∈ (Baseβ€˜πΎ)) β†’ ((0.β€˜πΎ) ∨ 𝑇) = 𝑇)
135, 9, 12syl2anc 583 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((0.β€˜πΎ) ∨ 𝑇) = 𝑇)
14 simp23 1206 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑆 = (0.β€˜πΎ))
1514oveq1d 7429 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝑆 ∨ 𝑇) = ((0.β€˜πΎ) ∨ 𝑇))
1610, 7hlatjidm 38778 . . . 4 ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴) β†’ (𝑇 ∨ 𝑇) = 𝑇)
173, 1, 16syl2anc 583 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝑇 ∨ 𝑇) = 𝑇)
1813, 15, 173eqtr4d 2777 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝑆 ∨ 𝑇) = (𝑇 ∨ 𝑇))
19 breq1 5145 . . . . 5 (𝑧 = 𝑇 β†’ (𝑧 ≀ π‘Š ↔ 𝑇 ≀ π‘Š))
2019notbid 318 . . . 4 (𝑧 = 𝑇 β†’ (Β¬ 𝑧 ≀ π‘Š ↔ Β¬ 𝑇 ≀ π‘Š))
21 oveq2 7422 . . . . 5 (𝑧 = 𝑇 β†’ (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑇))
22 oveq2 7422 . . . . 5 (𝑧 = 𝑇 β†’ (𝑇 ∨ 𝑧) = (𝑇 ∨ 𝑇))
2321, 22eqeq12d 2743 . . . 4 (𝑧 = 𝑇 β†’ ((𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧) ↔ (𝑆 ∨ 𝑇) = (𝑇 ∨ 𝑇)))
2420, 23anbi12d 630 . . 3 (𝑧 = 𝑇 β†’ ((Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)) ↔ (Β¬ 𝑇 ≀ π‘Š ∧ (𝑆 ∨ 𝑇) = (𝑇 ∨ 𝑇))))
2524rspcev 3607 . 2 ((𝑇 ∈ 𝐴 ∧ (Β¬ 𝑇 ≀ π‘Š ∧ (𝑆 ∨ 𝑇) = (𝑇 ∨ 𝑇))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
261, 2, 18, 25syl12anc 836 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 = (0.β€˜πΎ)) ∧ (𝑃 β‰  𝑄 ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑆 ∨ 𝑧) = (𝑇 ∨ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆƒwrex 3065   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  Basecbs 17171  lecple 17231  joincjn 18294  0.cp0 18406  OLcol 38583  Atomscatm 38672  HLchlt 38759  LHypclh 39394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-proset 18278  df-poset 18296  df-lub 18329  df-glb 18330  df-join 18331  df-meet 18332  df-p0 18408  df-lat 18415  df-oposet 38585  df-ol 38587  df-oml 38588  df-ats 38676  df-atl 38707  df-cvlat 38731  df-hlat 38760
This theorem is referenced by:  4atex2-0bOLDN  39489
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