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Theorem osumcllem6N 39136
Description: Lemma for osumclN 39142. Use atom exchange hlatexch1 38570 to swap 𝑝 and π‘ž. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
osumcllem.l ≀ = (leβ€˜πΎ)
osumcllem.j ∨ = (joinβ€˜πΎ)
osumcllem.a 𝐴 = (Atomsβ€˜πΎ)
osumcllem.p + = (+π‘ƒβ€˜πΎ)
osumcllem.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
osumcllem.c 𝐢 = (PSubClβ€˜πΎ)
osumcllem.m 𝑀 = (𝑋 + {𝑝})
osumcllem.u π‘ˆ = ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ)))
Assertion
Ref Expression
osumcllem6N (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ 𝑝 ∈ (𝑋 + π‘Œ))
Proof of Theorem osumcllem6N
StepHypRef Expression
1 simp11 1202 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ 𝐾 ∈ HL)
2 simp12 1203 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ 𝑋 βŠ† 𝐴)
3 simp13 1204 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ π‘Œ βŠ† 𝐴)
4 simp2r 1199 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ 𝑝 ∈ 𝐴)
5 simp31 1208 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ π‘Ÿ ∈ 𝑋)
6 simp32 1209 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ π‘ž ∈ π‘Œ)
73, 6sseldd 3983 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ π‘ž ∈ 𝐴)
82, 5sseldd 3983 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ π‘Ÿ ∈ 𝐴)
97, 4, 83jca 1127 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ (π‘ž ∈ 𝐴 ∧ 𝑝 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴))
10 simp2l 1198 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ))
11 osumcllem.l . . . . . 6 ≀ = (leβ€˜πΎ)
12 osumcllem.j . . . . . 6 ∨ = (joinβ€˜πΎ)
13 osumcllem.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
14 osumcllem.p . . . . . 6 + = (+π‘ƒβ€˜πΎ)
15 osumcllem.o . . . . . 6 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
16 osumcllem.c . . . . . 6 𝐢 = (PSubClβ€˜πΎ)
17 osumcllem.m . . . . . 6 𝑀 = (𝑋 + {𝑝})
18 osumcllem.u . . . . . 6 π‘ˆ = ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ)))
1911, 12, 13, 14, 15, 16, 17, 18osumcllem4N 39134 . . . . 5 (((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ)) β†’ π‘ž β‰  π‘Ÿ)
201, 3, 10, 5, 6, 19syl32anc 1377 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ π‘ž β‰  π‘Ÿ)
211, 9, 203jca 1127 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ (𝐾 ∈ HL ∧ (π‘ž ∈ 𝐴 ∧ 𝑝 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ π‘ž β‰  π‘Ÿ))
22 simp33 1210 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ π‘ž ≀ (π‘Ÿ ∨ 𝑝))
2311, 12, 13hlatexch1 38570 . . 3 ((𝐾 ∈ HL ∧ (π‘ž ∈ 𝐴 ∧ 𝑝 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ π‘ž β‰  π‘Ÿ) β†’ (π‘ž ≀ (π‘Ÿ ∨ 𝑝) β†’ 𝑝 ≀ (π‘Ÿ ∨ π‘ž)))
2421, 22, 23sylc 65 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ 𝑝 ≀ (π‘Ÿ ∨ π‘ž))
2511, 12, 13, 14, 15, 16, 17, 18osumcllem5N 39135 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ 𝑝 ≀ (π‘Ÿ ∨ π‘ž))) β†’ 𝑝 ∈ (𝑋 + π‘Œ))
261, 2, 3, 4, 5, 6, 24, 25syl313anc 1393 1 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑝 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝑋 ∧ π‘ž ∈ π‘Œ ∧ π‘ž ≀ (π‘Ÿ ∨ 𝑝))) β†’ 𝑝 ∈ (𝑋 + π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939   βŠ† wss 3948  {csn 4628   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  lecple 17209  joincjn 18269  Atomscatm 38437  HLchlt 38524  +𝑃cpadd 38970  βŠ₯𝑃cpolN 39077  PSubClcpscN 39109
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-pmap 38679  df-padd 38971  df-polarityN 39078
This theorem is referenced by:  osumcllem7N  39137
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