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Theorem cdlemefr29exN 36007
 Description: Lemma for cdlemefs29bpre1N 36022. (Compare cdleme25a 35958.) 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 1111 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp2r 1108 . . 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 35628 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋))
101, 2, 9syl2anc 694 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → ∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋))
11 nfv 1883 . . . 4 𝑠((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
12 nfv 1883 . . . 4 𝑠(𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
13 nfra1 2970 . . . 4 𝑠𝑠𝐴 𝐶𝐵
1411, 12, 13nf3an 1871 . . 3 𝑠(((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵)
15 simp11l 1192 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝐾 ∈ HL)
1615adantr 480 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐾 ∈ HL)
17 hllat 34968 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1816, 17syl 17 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐾 ∈ Lat)
19 simpl3 1086 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → ∀𝑠𝐴 𝐶𝐵)
20 simprl 809 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑠𝐴)
21 rsp 2958 . . . . . . . . 9 (∀𝑠𝐴 𝐶𝐵 → (𝑠𝐴𝐶𝐵))
2219, 20, 21sylc 65 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐶𝐵)
2315, 17syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝐾 ∈ Lat)
24 simp2rl 1150 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝑋𝐵)
25 simp11r 1193 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝑊𝐻)
263, 8lhpbase 35602 . . . . . . . . . . 11 (𝑊𝐻𝑊𝐵)
2725, 26syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → 𝑊𝐵)
283, 6latmcl 17099 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
2923, 24, 27, 28syl3anc 1366 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) → (𝑋 𝑊) ∈ 𝐵)
3029adantr 480 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑋 𝑊) ∈ 𝐵)
313, 5latjcl 17098 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝐶𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝐶 (𝑋 𝑊)) ∈ 𝐵)
3218, 22, 30, 31syl3anc 1366 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝐶 (𝑋 𝑊)) ∈ 𝐵)
3332expr 642 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ∀𝑠𝐴 𝐶𝐵) ∧ 𝑠𝐴) → (¬ 𝑠 𝑊 → (𝐶 (𝑋 𝑊)) ∈ 𝐵))
3433adantrd 483 . . . . 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))
3534ancld 575 . . . 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)))
3635ex 449 . . 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))))
3714, 36reximdai 3041 . 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)))
3810, 37mpd 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 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Latclat 17092  Atomscatm 34868  HLchlt 34955  LHypclh 35588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-lhyp 35592 This theorem is referenced by: (None)
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