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Theorem cdleme0cq 38681
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)
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
cdleme0cq (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ (𝑄 ∨ π‘ˆ) = (𝑃 ∨ 𝑄))

Proof of Theorem cdleme0cq
StepHypRef Expression
1 cdleme0.u . . 3 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
21oveq2i 7369 . 2 (𝑄 ∨ π‘ˆ) = (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))
3 simpll 766 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ 𝐾 ∈ HL)
4 simprrl 780 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ 𝑄 ∈ 𝐴)
5 hllat 37828 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
65ad2antrr 725 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ 𝐾 ∈ Lat)
7 eqid 2737 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
8 cdleme0.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
97, 8atbase 37754 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
109ad2antrl 727 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
117, 8atbase 37754 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
124, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
13 cdleme0.j . . . . . 6 ∨ = (joinβ€˜πΎ)
147, 13latjcl 18329 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
156, 10, 12, 14syl3anc 1372 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
16 cdleme0.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
177, 16lhpbase 38464 . . . . 5 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
1817ad2antlr 726 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ π‘Š ∈ (Baseβ€˜πΎ))
19 cdleme0.l . . . . . 6 ≀ = (leβ€˜πΎ)
207, 19, 13latlej2 18339 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ 𝑄 ≀ (𝑃 ∨ 𝑄))
216, 10, 12, 20syl3anc 1372 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ 𝑄 ≀ (𝑃 ∨ 𝑄))
22 cdleme0.m . . . . 5 ∧ = (meetβ€˜πΎ)
237, 19, 13, 22, 8atmod3i1 38330 . . . 4 ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) ∧ 𝑄 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ π‘Š)))
243, 4, 15, 18, 21, 23syl131anc 1384 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ π‘Š)))
25 eqid 2737 . . . . . 6 (1.β€˜πΎ) = (1.β€˜πΎ)
2619, 13, 25, 8, 16lhpjat2 38487 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (𝑄 ∨ π‘Š) = (1.β€˜πΎ))
2726adantrl 715 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ (𝑄 ∨ π‘Š) = (1.β€˜πΎ))
2827oveq2d 7374 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ π‘Š)) = ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)))
29 hlol 37826 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
3029ad2antrr 725 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ 𝐾 ∈ OL)
317, 22, 25olm11 37692 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)) = (𝑃 ∨ 𝑄))
3230, 15, 31syl2anc 585 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)) = (𝑃 ∨ 𝑄))
3324, 28, 323eqtrd 2781 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = (𝑃 ∨ 𝑄))
342, 33eqtrid 2789 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ (𝑄 ∨ π‘ˆ) = (𝑃 ∨ 𝑄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  joincjn 18201  meetcmee 18202  1.cp1 18314  Latclat 18321  OLcol 37639  Atomscatm 37728  HLchlt 37815  LHypclh 38450
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-clat 18389  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-psubsp 37969  df-pmap 37970  df-padd 38262  df-lhyp 38454
This theorem is referenced by:  cdleme11g  38731  cdlemg4b2  39076  cdlemg13a  39117
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