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Theorem cdlemg2jOLDN 39469
Description: TODO: Replace this with ltrnj 39003. (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg2inv.h 𝐻 = (LHypβ€˜πΎ)
cdlemg2inv.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemg2j.l ≀ = (leβ€˜πΎ)
cdlemg2j.j ∨ = (joinβ€˜πΎ)
cdlemg2j.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cdlemg2jOLDN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) β†’ (πΉβ€˜(𝑃 ∨ 𝑄)) = ((πΉβ€˜π‘ƒ) ∨ (πΉβ€˜π‘„)))

Proof of Theorem cdlemg2jOLDN
Dummy variables π‘ž 𝑝 𝑠 𝑑 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . 2 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 cdlemg2j.l . 2 ≀ = (leβ€˜πΎ)
3 cdlemg2j.j . 2 ∨ = (joinβ€˜πΎ)
4 eqid 2733 . 2 (meetβ€˜πΎ) = (meetβ€˜πΎ)
5 cdlemg2j.a . 2 𝐴 = (Atomsβ€˜πΎ)
6 cdlemg2inv.h . 2 𝐻 = (LHypβ€˜πΎ)
7 cdlemg2inv.t . 2 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
8 eqid 2733 . 2 ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š) = ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š)
9 eqid 2733 . 2 ((𝑑 ∨ ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(π‘ž ∨ ((𝑝 ∨ 𝑑)(meetβ€˜πΎ)π‘Š))) = ((𝑑 ∨ ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(π‘ž ∨ ((𝑝 ∨ 𝑑)(meetβ€˜πΎ)π‘Š)))
10 eqid 2733 . 2 ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)(((𝑑 ∨ ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(π‘ž ∨ ((𝑝 ∨ 𝑑)(meetβ€˜πΎ)π‘Š))) ∨ ((𝑠 ∨ 𝑑)(meetβ€˜πΎ)π‘Š))) = ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)(((𝑑 ∨ ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(π‘ž ∨ ((𝑝 ∨ 𝑑)(meetβ€˜πΎ)π‘Š))) ∨ ((𝑠 ∨ 𝑑)(meetβ€˜πΎ)π‘Š)))
11 eqid 2733 . 2 (π‘₯ ∈ (Baseβ€˜πΎ) ↦ if((𝑝 β‰  π‘ž ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ (Baseβ€˜πΎ)βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑝 ∨ π‘ž), (℩𝑦 ∈ (Baseβ€˜πΎ)βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑝 ∨ π‘ž)) β†’ 𝑦 = ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)(((𝑑 ∨ ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(π‘ž ∨ ((𝑝 ∨ 𝑑)(meetβ€˜πΎ)π‘Š))) ∨ ((𝑠 ∨ 𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑 ∨ ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(π‘ž ∨ ((𝑝 ∨ 𝑑)(meetβ€˜πΎ)π‘Š)))) ∨ (π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯)) = (π‘₯ ∈ (Baseβ€˜πΎ) ↦ if((𝑝 β‰  π‘ž ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ (Baseβ€˜πΎ)βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑝 ∨ π‘ž), (℩𝑦 ∈ (Baseβ€˜πΎ)βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑝 ∨ π‘ž)) β†’ 𝑦 = ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)(((𝑑 ∨ ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(π‘ž ∨ ((𝑝 ∨ 𝑑)(meetβ€˜πΎ)π‘Š))) ∨ ((𝑠 ∨ 𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑 ∨ ((𝑝 ∨ π‘ž)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(π‘ž ∨ ((𝑝 ∨ 𝑑)(meetβ€˜πΎ)π‘Š)))) ∨ (π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdlemg2jlemOLDN 39464 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) β†’ (πΉβ€˜(𝑃 ∨ 𝑄)) = ((πΉβ€˜π‘ƒ) ∨ (πΉβ€˜π‘„)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  β¦‹csb 3894  ifcif 4529   class class class wbr 5149   ↦ cmpt 5232  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  meetcmee 18265  Atomscatm 38133  HLchlt 38220  LHypclh 38855  LTrncltrn 38972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-riotaBAD 37823
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-undef 8258  df-map 8822  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lvols 38371  df-lines 38372  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030
This theorem is referenced by: (None)
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