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Theorem pexmidlem6N 38938
Description: Lemma for pexmidN 38932. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidlem.l ≀ = (leβ€˜πΎ)
pexmidlem.j ∨ = (joinβ€˜πΎ)
pexmidlem.a 𝐴 = (Atomsβ€˜πΎ)
pexmidlem.p + = (+π‘ƒβ€˜πΎ)
pexmidlem.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
pexmidlem.m 𝑀 = (𝑋 + {𝑝})
Assertion
Ref Expression
pexmidlem6N (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑀 = 𝑋)

Proof of Theorem pexmidlem6N
StepHypRef Expression
1 pexmidlem.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
2 pexmidlem.j . . . . . . . 8 ∨ = (joinβ€˜πΎ)
3 pexmidlem.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
4 pexmidlem.p . . . . . . . 8 + = (+π‘ƒβ€˜πΎ)
5 pexmidlem.o . . . . . . . 8 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
6 pexmidlem.m . . . . . . . 8 𝑀 = (𝑋 + {𝑝})
71, 2, 3, 4, 5, 6pexmidlem5N 38937 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (( βŠ₯ β€˜π‘‹) ∩ 𝑀) = βˆ…)
873adantr1 1169 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (( βŠ₯ β€˜π‘‹) ∩ 𝑀) = βˆ…)
98fveq2d 6895 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ 𝑀)) = ( βŠ₯ β€˜βˆ…))
10 simpl1 1191 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝐾 ∈ HL)
113, 5pol0N 38872 . . . . . 6 (𝐾 ∈ HL β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
1210, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
139, 12eqtrd 2772 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ 𝑀)) = 𝐴)
1413ineq1d 4211 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ 𝑀)) ∩ 𝑀) = (𝐴 ∩ 𝑀))
15 simpl2 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑋 βŠ† 𝐴)
16 simpl3 1193 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑝 ∈ 𝐴)
1716snssd 4812 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ {𝑝} βŠ† 𝐴)
183, 4paddssat 38777 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ {𝑝} βŠ† 𝐴) β†’ (𝑋 + {𝑝}) βŠ† 𝐴)
1910, 15, 17, 18syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 + {𝑝}) βŠ† 𝐴)
206, 19eqsstrid 4030 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑀 βŠ† 𝐴)
2110, 15, 203jca 1128 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑀 βŠ† 𝐴))
223, 4sspadd1 38778 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ {𝑝} βŠ† 𝐴) β†’ 𝑋 βŠ† (𝑋 + {𝑝}))
2310, 15, 17, 22syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑋 βŠ† (𝑋 + {𝑝}))
2423, 6sseqtrrdi 4033 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑋 βŠ† 𝑀)
25 simpr1 1194 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
26 eqid 2732 . . . . . . . . . . 11 (PSubClβ€˜πΎ) = (PSubClβ€˜πΎ)
273, 5, 26ispsubclN 38900 . . . . . . . . . 10 (𝐾 ∈ HL β†’ (𝑋 ∈ (PSubClβ€˜πΎ) ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
2810, 27syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 ∈ (PSubClβ€˜πΎ) ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
2915, 25, 28mpbir2and 711 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑋 ∈ (PSubClβ€˜πΎ))
303, 4, 26paddatclN 38912 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubClβ€˜πΎ) ∧ 𝑝 ∈ 𝐴) β†’ (𝑋 + {𝑝}) ∈ (PSubClβ€˜πΎ))
3110, 29, 16, 30syl3anc 1371 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 + {𝑝}) ∈ (PSubClβ€˜πΎ))
326, 31eqeltrid 2837 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑀 ∈ (PSubClβ€˜πΎ))
335, 26psubcli2N 38902 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑀 ∈ (PSubClβ€˜πΎ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘€)) = 𝑀)
3410, 32, 33syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘€)) = 𝑀)
3524, 34jca 512 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 βŠ† 𝑀 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘€)) = 𝑀))
363, 5poml4N 38916 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑀 βŠ† 𝐴) β†’ ((𝑋 βŠ† 𝑀 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘€)) = 𝑀) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ 𝑀)) ∩ 𝑀) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))))
3721, 35, 36sylc 65 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ 𝑀)) ∩ 𝑀) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
38 sseqin2 4215 . . . 4 (𝑀 βŠ† 𝐴 ↔ (𝐴 ∩ 𝑀) = 𝑀)
3920, 38sylib 217 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝐴 ∩ 𝑀) = 𝑀)
4014, 37, 393eqtr3rd 2781 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑀 = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
4140, 25eqtrd 2772 1 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑀 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  β€˜cfv 6543  (class class class)co 7411  lecple 17206  joincjn 18266  Atomscatm 38225  HLchlt 38312  +𝑃cpadd 38758  βŠ₯𝑃cpolN 38865  PSubClcpscN 38897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18387  df-clat 18454  df-oposet 38138  df-ol 38140  df-oml 38141  df-covers 38228  df-ats 38229  df-atl 38260  df-cvlat 38284  df-hlat 38313  df-psubsp 38466  df-pmap 38467  df-padd 38759  df-polarityN 38866  df-psubclN 38898
This theorem is referenced by:  pexmidlem8N  38940
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