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Theorem cdleme8 39755
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. 𝐢 represents s1. In their notation, we prove p ∨ s1 = p ∨ s. (Contributed by NM, 9-Jun-2012.)
Hypotheses
Ref Expression
cdleme8.l ≀ = (leβ€˜πΎ)
cdleme8.j ∨ = (joinβ€˜πΎ)
cdleme8.m ∧ = (meetβ€˜πΎ)
cdleme8.a 𝐴 = (Atomsβ€˜πΎ)
cdleme8.h 𝐻 = (LHypβ€˜πΎ)
cdleme8.4 𝐢 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
Assertion
Ref Expression
cdleme8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝐢) = (𝑃 ∨ 𝑆))

Proof of Theorem cdleme8
StepHypRef Expression
1 cdleme8.4 . . 3 𝐢 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
21oveq2i 7437 . 2 (𝑃 ∨ 𝐢) = (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š))
3 simp1l 1194 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ 𝐾 ∈ HL)
4 simp2l 1196 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ 𝑃 ∈ 𝐴)
53hllatd 38868 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ 𝐾 ∈ Lat)
6 eqid 2728 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
7 cdleme8.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
86, 7atbase 38793 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
94, 8syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
106, 7atbase 38793 . . . . . 6 (𝑆 ∈ 𝐴 β†’ 𝑆 ∈ (Baseβ€˜πΎ))
11103ad2ant3 1132 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ 𝑆 ∈ (Baseβ€˜πΎ))
12 cdleme8.j . . . . . 6 ∨ = (joinβ€˜πΎ)
136, 12latjcl 18438 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑆 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
145, 9, 11, 13syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
15 simp1r 1195 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ π‘Š ∈ 𝐻)
16 cdleme8.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
176, 16lhpbase 39503 . . . . 5 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
1815, 17syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ π‘Š ∈ (Baseβ€˜πΎ))
19 cdleme8.l . . . . . 6 ≀ = (leβ€˜πΎ)
206, 19, 12latlej1 18447 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑆 ∈ (Baseβ€˜πΎ)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑆))
215, 9, 11, 20syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ 𝑃 ≀ (𝑃 ∨ 𝑆))
22 cdleme8.m . . . . 5 ∧ = (meetβ€˜πΎ)
236, 19, 12, 22, 7atmod3i1 39369 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) ∧ 𝑃 ≀ (𝑃 ∨ 𝑆)) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)) = ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ π‘Š)))
243, 4, 14, 18, 21, 23syl131anc 1380 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)) = ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ π‘Š)))
25 eqid 2728 . . . . . 6 (1.β€˜πΎ) = (1.β€˜πΎ)
2619, 12, 25, 7, 16lhpjat2 39526 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (𝑃 ∨ π‘Š) = (1.β€˜πΎ))
27263adant3 1129 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ π‘Š) = (1.β€˜πΎ))
2827oveq2d 7442 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ π‘Š)) = ((𝑃 ∨ 𝑆) ∧ (1.β€˜πΎ)))
29 hlol 38865 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
303, 29syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ 𝐾 ∈ OL)
316, 22, 25olm11 38731 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑆) ∧ (1.β€˜πΎ)) = (𝑃 ∨ 𝑆))
3230, 14, 31syl2anc 582 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ ((𝑃 ∨ 𝑆) ∧ (1.β€˜πΎ)) = (𝑃 ∨ 𝑆))
3324, 28, 323eqtrd 2772 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)) = (𝑃 ∨ 𝑆))
342, 33eqtrid 2780 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝐢) = (𝑃 ∨ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  joincjn 18310  meetcmee 18311  1.cp1 18423  Latclat 18430  OLcol 38678  Atomscatm 38767  HLchlt 38854  LHypclh 39489
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493
This theorem is referenced by:  cdleme8tN  39760  cdleme15a  39779  cdleme17b  39792  cdlemg3a  40102
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