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Theorem cdlemefr29exN 40112
Description: Lemma for cdlemefs29bpre1N 40127. (Compare cdleme25a 40063.) TODO: FIX COMMENT. TODO: IS THIS NEEDED? (Contributed by NM, 28-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemefr29.b 𝐵 = (Base‘𝐾)
cdlemefr29.l = (le‘𝐾)
cdlemefr29.j = (join‘𝐾)
cdlemefr29.m = (meet‘𝐾)
cdlemefr29.a 𝐴 = (Atoms‘𝐾)
cdlemefr29.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
cdlemefr29exN ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → ∃𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵))
Distinct variable groups:   𝐴,𝑠   𝐵,𝑠   𝐻,𝑠   𝐾,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑊,𝑠   𝑋,𝑠
Allowed substitution hints:   𝐶(𝑠)   (𝑠)

Proof of Theorem cdlemefr29exN
StepHypRef Expression
1 simp11 1200 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp2r 1197 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
3 cdlemefr29.b . . . 4 𝐵 = (Base‘𝐾)
4 cdlemefr29.l . . . 4 = (le‘𝐾)
5 cdlemefr29.j . . . 4 = (join‘𝐾)
6 cdlemefr29.m . . . 4 = (meet‘𝐾)
7 cdlemefr29.a . . . 4 𝐴 = (Atoms‘𝐾)
8 cdlemefr29.h . . . 4 𝐻 = (LHyp‘𝐾)
93, 4, 5, 6, 7, 8lhpmcvr2 39734 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋))
101, 2, 9syl2anc 582 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → ∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋))
11 nfv 1910 . . . 4 𝑠((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
12 nfv 1910 . . . 4 𝑠(𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
13 nfra1 3272 . . . 4 𝑠𝑠𝐴 𝐶𝐵
1411, 12, 13nf3an 1897 . . 3 𝑠(((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵)
15 simp11l 1281 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝐾 ∈ HL)
1615adantr 479 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐾 ∈ HL)
1716hllatd 39073 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐾 ∈ Lat)
18 simpl3 1190 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → ∀𝑠𝐴 𝐶𝐵)
19 simprl 769 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑠𝐴)
20 rsp 3235 . . . . . . . . 9 (∀𝑠𝐴 𝐶𝐵 → (𝑠𝐴𝐶𝐵))
2118, 19, 20sylc 65 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐶𝐵)
2215hllatd 39073 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝐾 ∈ Lat)
23 simp2rl 1239 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝑋𝐵)
24 simp11r 1282 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝑊𝐻)
253, 8lhpbase 39708 . . . . . . . . . . 11 (𝑊𝐻𝑊𝐵)
2624, 25syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝑊𝐵)
273, 6latmcl 18458 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
2822, 23, 26, 27syl3anc 1368 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → (𝑋 𝑊) ∈ 𝐵)
2928adantr 479 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑋 𝑊) ∈ 𝐵)
303, 5latjcl 18457 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝐶𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝐶 (𝑋 𝑊)) ∈ 𝐵)
3117, 21, 29, 30syl3anc 1368 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝐶 (𝑋 𝑊)) ∈ 𝐵)
3231expr 455 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ 𝑠𝐴) → (¬ 𝑠 𝑊 → (𝐶 (𝑋 𝑊)) ∈ 𝐵))
3332adantrd 490 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ 𝑠𝐴) → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → (𝐶 (𝑋 𝑊)) ∈ 𝐵))
3433ancld 549 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ 𝑠𝐴) → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵)))
3534ex 411 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → (𝑠𝐴 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵))))
3614, 35reximdai 3249 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → (∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → ∃𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵)))
3710, 36mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → ∃𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060   class class class wbr 5144  cfv 6544  (class class class)co 7414  Basecbs 17206  lecple 17266  joincjn 18329  meetcmee 18330  Latclat 18449  Atomscatm 38972  HLchlt 39059  LHypclh 39694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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 7370  df-ov 7417  df-oprab 7418  df-proset 18313  df-poset 18331  df-plt 18348  df-lub 18364  df-glb 18365  df-join 18366  df-meet 18367  df-p0 18443  df-p1 18444  df-lat 18450  df-clat 18517  df-oposet 38885  df-ol 38887  df-oml 38888  df-covers 38975  df-ats 38976  df-atl 39007  df-cvlat 39031  df-hlat 39060  df-lhyp 39698
This theorem is referenced by: (None)
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