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Theorem cdleme29b 34479
Description: Transform cdleme28 34477. (Compare cdleme25b 34458.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b 𝐵 = (Base‘𝐾)
cdleme26.l = (le‘𝐾)
cdleme26.j = (join‘𝐾)
cdleme26.m = (meet‘𝐾)
cdleme26.a 𝐴 = (Atoms‘𝐾)
cdleme26.h 𝐻 = (LHyp‘𝐾)
cdleme27.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme27.f 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme27.z 𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
cdleme27.n 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
cdleme27.d 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
cdleme27.c 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
Assertion
Ref Expression
cdleme29b ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑣𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑣 = (𝐶 (𝑋 𝑊))))
Distinct variable groups:   𝑢,𝑠,𝑧,𝐴   𝐵,𝑠,𝑢,𝑧   𝑢,𝐹   𝐻,𝑠,𝑧   ,𝑠,𝑢,𝑧   𝐾,𝑠,𝑧   ,𝑠,𝑢,𝑧   ,𝑠,𝑢,𝑧   𝑢,𝑁   𝑃,𝑠,𝑢,𝑧   𝑄,𝑠,𝑢,𝑧   𝑈,𝑠,𝑢,𝑧   𝑊,𝑠,𝑢,𝑧   𝑋,𝑠   𝑣,𝐴   𝑣,𝐵   𝑣,   𝑣,   𝑣,   𝑣,𝑃   𝑣,𝑄   𝑣,𝑈   𝑣,𝑊   𝑣,𝐶   𝑣,𝑠,𝑍,𝑢   𝑧,𝑣,𝑋
Allowed substitution hints:   𝐶(𝑧,𝑢,𝑠)   𝐷(𝑧,𝑣,𝑢,𝑠)   𝐹(𝑧,𝑣,𝑠)   𝐻(𝑣,𝑢)   𝐾(𝑣,𝑢)   𝑁(𝑧,𝑣,𝑠)   𝑋(𝑢)   𝑍(𝑧)

Proof of Theorem cdleme29b
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cdleme26.b . . 3 𝐵 = (Base‘𝐾)
2 cdleme26.l . . 3 = (le‘𝐾)
3 cdleme26.j . . 3 = (join‘𝐾)
4 cdleme26.m . . 3 = (meet‘𝐾)
5 cdleme26.a . . 3 𝐴 = (Atoms‘𝐾)
6 cdleme26.h . . 3 𝐻 = (LHyp‘𝐾)
7 cdleme27.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
8 cdleme27.f . . 3 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
9 cdleme27.z . . 3 𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
10 cdleme27.n . . 3 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
11 cdleme27.d . . 3 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
12 cdleme27.c . . 3 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdleme29ex 34478 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵))
14 eqid 2604 . . 3 ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))) = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
15 eqid 2604 . . 3 ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))) = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
16 eqid 2604 . . 3 (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))) = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))))
17 eqid 2604 . . 3 if(𝑡 (𝑃 𝑄), (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))), ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))) = if(𝑡 (𝑃 𝑄), (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))), ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))))
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17cdleme28 34477 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∀𝑠𝐴𝑡𝐴 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (¬ 𝑡 𝑊 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → (𝐶 (𝑋 𝑊)) = (if(𝑡 (𝑃 𝑄), (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))), ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))) (𝑋 𝑊))))
19 breq1 4575 . . . . . 6 (𝑠 = 𝑡 → (𝑠 𝑊𝑡 𝑊))
2019notbid 306 . . . . 5 (𝑠 = 𝑡 → (¬ 𝑠 𝑊 ↔ ¬ 𝑡 𝑊))
21 oveq1 6529 . . . . . 6 (𝑠 = 𝑡 → (𝑠 (𝑋 𝑊)) = (𝑡 (𝑋 𝑊)))
2221eqeq1d 2606 . . . . 5 (𝑠 = 𝑡 → ((𝑠 (𝑋 𝑊)) = 𝑋 ↔ (𝑡 (𝑋 𝑊)) = 𝑋))
2320, 22anbi12d 742 . . . 4 (𝑠 = 𝑡 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ↔ (¬ 𝑡 𝑊 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)))
2412oveq1i 6532 . . . . 5 (𝐶 (𝑋 𝑊)) = (if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) (𝑋 𝑊))
25 breq1 4575 . . . . . . 7 (𝑠 = 𝑡 → (𝑠 (𝑃 𝑄) ↔ 𝑡 (𝑃 𝑄)))
26 oveq1 6529 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑡 → (𝑠 𝑧) = (𝑡 𝑧))
2726oveq1d 6537 . . . . . . . . . . . . . . 15 (𝑠 = 𝑡 → ((𝑠 𝑧) 𝑊) = ((𝑡 𝑧) 𝑊))
2827oveq2d 6538 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → (𝑍 ((𝑠 𝑧) 𝑊)) = (𝑍 ((𝑡 𝑧) 𝑊)))
2928oveq2d 6538 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊))) = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))
3010, 29syl5eq 2650 . . . . . . . . . . . 12 (𝑠 = 𝑡𝑁 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))
3130eqeq2d 2614 . . . . . . . . . . 11 (𝑠 = 𝑡 → (𝑢 = 𝑁𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))))
3231imbi2d 328 . . . . . . . . . 10 (𝑠 = 𝑡 → (((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))))
3332ralbidv 2963 . . . . . . . . 9 (𝑠 = 𝑡 → (∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))))
3433riotabidv 6486 . . . . . . . 8 (𝑠 = 𝑡 → (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁)) = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))))
3511, 34syl5eq 2650 . . . . . . 7 (𝑠 = 𝑡𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))))
36 oveq1 6529 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑠 𝑈) = (𝑡 𝑈))
37 oveq2 6530 . . . . . . . . . . 11 (𝑠 = 𝑡 → (𝑃 𝑠) = (𝑃 𝑡))
3837oveq1d 6537 . . . . . . . . . 10 (𝑠 = 𝑡 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑡) 𝑊))
3938oveq2d 6538 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑡) 𝑊)))
4036, 39oveq12d 6540 . . . . . . . 8 (𝑠 = 𝑡 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))))
418, 40syl5eq 2650 . . . . . . 7 (𝑠 = 𝑡𝐹 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))))
4225, 35, 41ifbieq12d 4057 . . . . . 6 (𝑠 = 𝑡 → if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) = if(𝑡 (𝑃 𝑄), (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))), ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))))
4342oveq1d 6537 . . . . 5 (𝑠 = 𝑡 → (if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) (𝑋 𝑊)) = (if(𝑡 (𝑃 𝑄), (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))), ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))) (𝑋 𝑊)))
4424, 43syl5eq 2650 . . . 4 (𝑠 = 𝑡 → (𝐶 (𝑋 𝑊)) = (if(𝑡 (𝑃 𝑄), (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))), ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))) (𝑋 𝑊)))
4523, 44reusv3 4792 . . 3 (∃𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵) → (∀𝑠𝐴𝑡𝐴 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (¬ 𝑡 𝑊 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → (𝐶 (𝑋 𝑊)) = (if(𝑡 (𝑃 𝑄), (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))), ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))) (𝑋 𝑊))) ↔ ∃𝑣𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑣 = (𝐶 (𝑋 𝑊)))))
4645biimpd 217 . 2 (∃𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝐶 (𝑋 𝑊)) ∈ 𝐵) → (∀𝑠𝐴𝑡𝐴 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (¬ 𝑡 𝑊 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → (𝐶 (𝑋 𝑊)) = (if(𝑡 (𝑃 𝑄), (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))), ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))) (𝑋 𝑊))) → ∃𝑣𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑣 = (𝐶 (𝑋 𝑊)))))
4713, 18, 46sylc 62 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑣𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑣 = (𝐶 (𝑋 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774  wral 2890  wrex 2891  ifcif 4030   class class class wbr 4572  cfv 5785  crio 6483  (class class class)co 6522  Basecbs 15636  lecple 15716  joincjn 16708  meetcmee 16709  Atomscatm 33366  HLchlt 33453  LHypclh 34086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-riotaBAD 33055
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-iin 4447  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-1st 7031  df-2nd 7032  df-undef 7258  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-p1 16804  df-lat 16810  df-clat 16872  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-llines 33600  df-lplanes 33601  df-lvols 33602  df-lines 33603  df-psubsp 33605  df-pmap 33606  df-padd 33898  df-lhyp 34090
This theorem is referenced by:  cdleme29c  34480
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