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Theorem pexmidlem6N 38846
Description: Lemma for pexmidN 38840. (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 38845 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (( βŠ₯ β€˜π‘‹) ∩ 𝑀) = βˆ…)
873adantr1 1170 . . . . . 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 6896 . . . . 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 1192 . . . . . 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 38780 . . . . . 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 2773 . . . 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 4212 . . 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 1193 . . . . 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 1194 . . . . . . . 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 4813 . . . . . . 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 38685 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ {𝑝} βŠ† 𝐴) β†’ (𝑋 + {𝑝}) βŠ† 𝐴)
1910, 15, 17, 18syl3anc 1372 . . . . . 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 4031 . . . . 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 1129 . . . 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 38686 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ {𝑝} βŠ† 𝐴) β†’ 𝑋 βŠ† (𝑋 + {𝑝}))
2310, 15, 17, 22syl3anc 1372 . . . . . 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 4034 . . . . 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 1195 . . . . . . . . 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 2733 . . . . . . . . . . 11 (PSubClβ€˜πΎ) = (PSubClβ€˜πΎ)
273, 5, 26ispsubclN 38808 . . . . . . . . . 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 712 . . . . . . . 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 38820 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubClβ€˜πΎ) ∧ 𝑝 ∈ 𝐴) β†’ (𝑋 + {𝑝}) ∈ (PSubClβ€˜πΎ))
3110, 29, 16, 30syl3anc 1372 . . . . . . 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 2838 . . . . . 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 38810 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑀 ∈ (PSubClβ€˜πΎ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘€)) = 𝑀)
3410, 32, 33syl2anc 585 . . . . 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 513 . . . 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 38824 . . . 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 4216 . . . 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 2782 . 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 2773 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  β€˜cfv 6544  (class class class)co 7409  lecple 17204  joincjn 18264  Atomscatm 38133  HLchlt 38220  +𝑃cpadd 38666  βŠ₯𝑃cpolN 38773  PSubClcpscN 38805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-polarityN 38774  df-psubclN 38806
This theorem is referenced by:  pexmidlem8N  38848
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