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Theorem ltrniotacnvN 40105
Description: Version of cdleme51finvtrN 40083 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
ltrniotacnvN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ ◑𝐹 ∈ 𝑇)
Distinct variable groups:   𝐴,𝑓   𝑓,𝐻   𝑓,𝐾   ≀ ,𝑓   𝑃,𝑓   𝑄,𝑓   𝑇,𝑓   𝑓,π‘Š
Allowed substitution hint:   𝐹(𝑓)
Proof of Theorem ltrniotacnvN
Dummy variables 𝑠 𝑑 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . 2 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 ltrniotaval.l . 2 ≀ = (leβ€˜πΎ)
3 eqid 2725 . 2 (joinβ€˜πΎ) = (joinβ€˜πΎ)
4 eqid 2725 . 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 2725 . 2 ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š) = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š)
8 eqid 2725 . 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 2725 . 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 2725 . 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, 12cdlemg1finvtrlemN 40100 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ ◑𝐹 ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  β¦‹csb 3886  ifcif 4525   class class class wbr 5144   ↦ cmpt 5227  β—‘ccnv 5672  β€˜cfv 6543  β„©crio 7368  (class class class)co 7413  Basecbs 17174  lecple 17234  joincjn 18297  meetcmee 18298  Atomscatm 38787  HLchlt 38874  LHypclh 39509  LTrncltrn 39626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-riotaBAD 38477
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-undef 8272  df-map 8840  df-proset 18281  df-poset 18299  df-plt 18316  df-lub 18332  df-glb 18333  df-join 18334  df-meet 18335  df-p0 18411  df-p1 18412  df-lat 18418  df-clat 18485  df-oposet 38700  df-ol 38702  df-oml 38703  df-covers 38790  df-ats 38791  df-atl 38822  df-cvlat 38846  df-hlat 38875  df-llines 39023  df-lplanes 39024  df-lvols 39025  df-lines 39026  df-psubsp 39028  df-pmap 39029  df-padd 39321  df-lhyp 39513  df-laut 39514  df-ldil 39629  df-ltrn 39630  df-trl 39684
This theorem is referenced by: (None)
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