Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ltrniotacnvN Structured version   Visualization version   GIF version

Theorem ltrniotacnvN 38856
Description: Version of cdleme51finvtrN 38834 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 2736 . 2 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 ltrniotaval.l . 2 ≀ = (leβ€˜πΎ)
3 eqid 2736 . 2 (joinβ€˜πΎ) = (joinβ€˜πΎ)
4 eqid 2736 . 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 2736 . 2 ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š) = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š)
8 eqid 2736 . 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 2736 . 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 2736 . 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 38851 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 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2940  βˆ€wral 3061  β¦‹csb 3843  ifcif 4473   class class class wbr 5092   ↦ cmpt 5175  β—‘ccnv 5619  β€˜cfv 6479  β„©crio 7292  (class class class)co 7337  Basecbs 17009  lecple 17066  joincjn 18126  meetcmee 18127  Atomscatm 37538  HLchlt 37625  LHypclh 38260  LTrncltrn 38377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-riotaBAD 37228
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-undef 8159  df-map 8688  df-proset 18110  df-poset 18128  df-plt 18145  df-lub 18161  df-glb 18162  df-join 18163  df-meet 18164  df-p0 18240  df-p1 18241  df-lat 18247  df-clat 18314  df-oposet 37451  df-ol 37453  df-oml 37454  df-covers 37541  df-ats 37542  df-atl 37573  df-cvlat 37597  df-hlat 37626  df-llines 37774  df-lplanes 37775  df-lvols 37776  df-lines 37777  df-psubsp 37779  df-pmap 37780  df-padd 38072  df-lhyp 38264  df-laut 38265  df-ldil 38380  df-ltrn 38381  df-trl 38435
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator