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Theorem cdleme43cN 39666
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 last line: r v g(s) = r v v2. (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme43.b 𝐡 = (Baseβ€˜πΎ)
cdleme43.l ≀ = (leβ€˜πΎ)
cdleme43.j ∨ = (joinβ€˜πΎ)
cdleme43.m ∧ = (meetβ€˜πΎ)
cdleme43.a 𝐴 = (Atomsβ€˜πΎ)
cdleme43.h 𝐻 = (LHypβ€˜πΎ)
cdleme43.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme43.x 𝑋 = ((𝑄 ∨ 𝑃) ∧ π‘Š)
cdleme43.c 𝐢 = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))
cdleme43.f 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝐢 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))
cdleme43.d 𝐷 = ((𝑆 ∨ 𝑋) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ π‘Š)))
cdleme43.g 𝐺 = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ π‘Š)))
cdleme43.e 𝐸 = ((𝐷 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝐷) ∧ π‘Š)))
cdleme43.v 𝑉 = ((𝑍 ∨ 𝑆) ∧ π‘Š)
cdleme43.y π‘Œ = ((𝑅 ∨ 𝐷) ∧ π‘Š)
Assertion
Ref Expression
cdleme43cN ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑅 ∨ 𝐷) = (𝑅 ∨ π‘Œ))

Proof of Theorem cdleme43cN
StepHypRef Expression
1 simp11 1202 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simp22 1206 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))
3 simp1 1135 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)))
4 simp21 1205 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑃 β‰  𝑄)
5 simp23 1207 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š))
6 simp3 1137 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))
7 cdleme43.b . . . 4 𝐡 = (Baseβ€˜πΎ)
8 cdleme43.l . . . 4 ≀ = (leβ€˜πΎ)
9 cdleme43.j . . . 4 ∨ = (joinβ€˜πΎ)
10 cdleme43.m . . . 4 ∧ = (meetβ€˜πΎ)
11 cdleme43.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
12 cdleme43.h . . . 4 𝐻 = (LHypβ€˜πΎ)
13 cdleme43.u . . . 4 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
14 cdleme43.x . . . 4 𝑋 = ((𝑄 ∨ 𝑃) ∧ π‘Š)
15 cdleme43.c . . . 4 𝐢 = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))
16 cdleme43.f . . . 4 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝐢 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))
17 cdleme43.d . . . 4 𝐷 = ((𝑆 ∨ 𝑋) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ π‘Š)))
18 cdleme43.g . . . 4 𝐺 = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ π‘Š)))
19 cdleme43.e . . . 4 𝐸 = ((𝐷 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝐷) ∧ π‘Š)))
20 cdleme43.v . . . 4 𝑉 = ((𝑍 ∨ 𝑆) ∧ π‘Š)
21 cdleme43.y . . . 4 π‘Œ = ((𝑅 ∨ 𝐷) ∧ π‘Š)
227, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21cdleme43bN 39665 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ (𝐷 ∈ 𝐴 ∧ Β¬ 𝐷 ≀ π‘Š))
233, 4, 5, 6, 22syl121anc 1374 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ (𝐷 ∈ 𝐴 ∧ Β¬ 𝐷 ≀ π‘Š))
247, 8, 9, 10, 11, 12, 21cdleme42a 39646 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝐷 ∈ 𝐴 ∧ Β¬ 𝐷 ≀ π‘Š)) β†’ (𝑅 ∨ 𝐷) = (𝑅 ∨ π‘Œ))
251, 2, 23, 24syl3anc 1370 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑅 ∨ 𝐷) = (𝑅 ∨ π‘Œ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  meetcmee 18270  Atomscatm 38437  HLchlt 38524  LHypclh 39159
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 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-lines 38676  df-psubsp 38678  df-pmap 38679  df-padd 38971  df-lhyp 39163
This theorem is referenced by:  cdlemeg46rjgN  39697
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