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Theorem cdleme46f2g2 39668
Description: Conversion for 𝐺 to reuse 𝐹 theorems. TODO FIX COMMENT. TODO What other conversion theorems would be reused? e.g. cdlemeg46nlpq 39692? Find other hlatjcom 38542 uses giving 𝑄 ∨ 𝑃. (Contributed by NM, 1-Apr-2013.)
Hypotheses
Ref Expression
cdleme46fg.j ∨ = (joinβ€˜πΎ)
cdleme46fg.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cdleme46f2g2 ((((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†’ (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝑄 β‰  𝑃 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑃)))

Proof of Theorem cdleme46f2g2
StepHypRef Expression
1 simp11 1202 . . 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†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simp13 1204 . . 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†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
3 simp12 1203 . . 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†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
41, 2, 33jca 1127 . 2 ((((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†’ ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)))
5 simp2l 1198 . . . 4 ((((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†’ 𝑃 β‰  𝑄)
65necomd 2995 . . 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†’ 𝑄 β‰  𝑃)
7 simp2r 1199 . . 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†’ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š))
86, 7jca 511 . 2 ((((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†’ (𝑄 β‰  𝑃 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)))
9 simpl1l 1223 . . . . . 6 ((((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†’ 𝐾 ∈ HL)
10 simpl2l 1225 . . . . . 6 ((((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†’ 𝑃 ∈ 𝐴)
11 simpl3l 1227 . . . . . 6 ((((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†’ 𝑄 ∈ 𝐴)
12 cdleme46fg.j . . . . . . 7 ∨ = (joinβ€˜πΎ)
13 cdleme46fg.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
1412, 13hlatjcom 38542 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
159, 10, 11, 14syl3anc 1370 . . . . 5 ((((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†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
1615breq2d 5160 . . . 4 ((((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) ↔ 𝑆 ≀ (𝑄 ∨ 𝑃)))
1716notbid 318 . . 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†’ (Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ↔ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑃)))
1817biimp3a 1468 . 2 ((((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†’ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑃))
194, 8, 183jca 1127 1 ((((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†’ (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝑄 β‰  𝑃 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑃)))
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  joincjn 18269  Atomscatm 38437  HLchlt 38524
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-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-lub 18304  df-join 18306  df-lat 18390  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525
This theorem is referenced by:  cdlemeg47b  39683  cdlemeg46c  39688  cdlemeg46ngfr  39693  cdlemeg46nfgr  39694
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