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Theorem cdleme0cp 39743
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 40126- swap consequent equality; make antecedent use df-3an 1086. (Contributed by NM, 13-Jun-2012.)
Hypotheses
Ref Expression
cdleme0.l ≀ = (leβ€˜πΎ)
cdleme0.j ∨ = (joinβ€˜πΎ)
cdleme0.m ∧ = (meetβ€˜πΎ)
cdleme0.a 𝐴 = (Atomsβ€˜πΎ)
cdleme0.h 𝐻 = (LHypβ€˜πΎ)
cdleme0.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
Assertion
Ref Expression
cdleme0cp (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ π‘ˆ) = (𝑃 ∨ 𝑄))

Proof of Theorem cdleme0cp
StepHypRef Expression
1 cdleme0.u . . 3 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
21oveq2i 7427 . 2 (𝑃 ∨ π‘ˆ) = (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))
3 simpll 765 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
4 simprll 777 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
5 hllat 38891 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
65ad2antrr 724 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
7 eqid 2725 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
8 cdleme0.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
97, 8atbase 38817 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
104, 9syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
11 simprr 771 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
127, 8atbase 38817 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
1311, 12syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
14 cdleme0.j . . . . . 6 ∨ = (joinβ€˜πΎ)
157, 14latjcl 18430 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
166, 10, 13, 15syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
17 cdleme0.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
187, 17lhpbase 39527 . . . . 5 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
1918ad2antlr 725 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ π‘Š ∈ (Baseβ€˜πΎ))
20 cdleme0.l . . . . . 6 ≀ = (leβ€˜πΎ)
2120, 14, 8hlatlej1 38903 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
223, 4, 11, 21syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
23 cdleme0.m . . . . 5 ∧ = (meetβ€˜πΎ)
247, 20, 14, 23, 8atmod3i1 39393 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) ∧ 𝑃 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = ((𝑃 ∨ 𝑄) ∧ (𝑃 ∨ π‘Š)))
253, 4, 16, 19, 22, 24syl131anc 1380 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = ((𝑃 ∨ 𝑄) ∧ (𝑃 ∨ π‘Š)))
26 eqid 2725 . . . . . 6 (1.β€˜πΎ) = (1.β€˜πΎ)
2720, 14, 26, 8, 17lhpjat2 39550 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (𝑃 ∨ π‘Š) = (1.β€˜πΎ))
2827adantrr 715 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ π‘Š) = (1.β€˜πΎ))
2928oveq2d 7432 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑃 ∨ π‘Š)) = ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)))
30 hlol 38889 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
3130ad2antrr 724 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ OL)
327, 23, 26olm11 38755 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)) = (𝑃 ∨ 𝑄))
3331, 16, 32syl2anc 582 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)) = (𝑃 ∨ 𝑄))
3425, 29, 333eqtrd 2769 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = (𝑃 ∨ 𝑄))
352, 34eqtrid 2777 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ π‘ˆ) = (𝑃 ∨ 𝑄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179  lecple 17239  joincjn 18302  meetcmee 18303  1.cp1 18415  Latclat 18422  OLcol 38702  Atomscatm 38791  HLchlt 38878  LHypclh 39513
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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  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 5144  df-opab 5206  df-mpt 5227  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-proset 18286  df-poset 18304  df-plt 18321  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-p0 18416  df-p1 18417  df-lat 18423  df-clat 18490  df-oposet 38704  df-ol 38706  df-oml 38707  df-covers 38794  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879  df-psubsp 39032  df-pmap 39033  df-padd 39325  df-lhyp 39517
This theorem is referenced by:  cdleme11c  39790  cdlemg4b1  40138  cdlemg4g  40145  cdlemg13a  40180  cdlemg17a  40190  cdlemg17f  40195  cdlemg18b  40208  cdlemg18c  40209
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