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Theorem ltrniotafvawN 39988
Description: Version of cdleme46fvaw 39911 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrniotaval.l ≀ = (leβ€˜πΎ)
ltrniotaval.a 𝐴 = (Atomsβ€˜πΎ)
ltrniotaval.h 𝐻 = (LHypβ€˜πΎ)
ltrniotaval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
ltrniotaval.f 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)
Assertion
Ref Expression
ltrniotafvawN ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ ((πΉβ€˜π‘…) ∈ 𝐴 ∧ Β¬ (πΉβ€˜π‘…) ≀ π‘Š))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐻   𝑓,𝐾   ≀ ,𝑓   𝑃,𝑓   𝑄,𝑓   𝑇,𝑓   𝑓,π‘Š
Allowed substitution hints:   𝑅(𝑓)   𝐹(𝑓)
Proof of Theorem ltrniotafvawN
Dummy variables 𝑠 𝑑 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2727 . 2 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 ltrniotaval.l . 2 ≀ = (leβ€˜πΎ)
3 eqid 2727 . 2 (joinβ€˜πΎ) = (joinβ€˜πΎ)
4 eqid 2727 . 2 (meetβ€˜πΎ) = (meetβ€˜πΎ)
5 ltrniotaval.a . 2 𝐴 = (Atomsβ€˜πΎ)
6 ltrniotaval.h . 2 𝐻 = (LHypβ€˜πΎ)
7 eqid 2727 . 2 ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š) = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š)
8 eqid 2727 . 2 ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))) = ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))
9 eqid 2727 . 2 ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))) = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))
10 eqid 2727 . 2 (π‘₯ ∈ (Baseβ€˜πΎ) ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ (Baseβ€˜πΎ)βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ (Baseβ€˜πΎ)βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯)) = (π‘₯ ∈ (Baseβ€˜πΎ) ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ (Baseβ€˜πΎ)βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ (Baseβ€˜πΎ)βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯))
11 ltrniotaval.t . 2 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
12 ltrniotaval.f . 2 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemg1fvawlemN 39983 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ ((πΉβ€˜π‘…) ∈ 𝐴 ∧ Β¬ (πΉβ€˜π‘…) ≀ π‘Š))
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  βˆ€wral 3056  β¦‹csb 3889  ifcif 4524   class class class wbr 5142   ↦ cmpt 5225  β€˜cfv 6542  β„©crio 7369  (class class class)co 7414  Basecbs 17171  lecple 17231  joincjn 18294  meetcmee 18295  Atomscatm 38672  HLchlt 38759  LHypclh 39394  LTrncltrn 39511
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  ax-riotaBAD 38362
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  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-iin 4994  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-mpo 7419  df-1st 7987  df-2nd 7988  df-undef 8272  df-map 8838  df-proset 18278  df-poset 18296  df-plt 18313  df-lub 18329  df-glb 18330  df-join 18331  df-meet 18332  df-p0 18408  df-p1 18409  df-lat 18415  df-clat 18482  df-oposet 38585  df-ol 38587  df-oml 38588  df-covers 38675  df-ats 38676  df-atl 38707  df-cvlat 38731  df-hlat 38760  df-llines 38908  df-lplanes 38909  df-lvols 38910  df-lines 38911  df-psubsp 38913  df-pmap 38914  df-padd 39206  df-lhyp 39398  df-laut 39399  df-ldil 39514  df-ltrn 39515  df-trl 39569
This theorem is referenced by: (None)
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