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Theorem cdleme29b 38012
Description: Transform cdleme28 38010. (Compare cdleme25b 37991.) 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 38011 . 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 2738 . . 3 ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))) = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
15 eqid 2738 . . 3 ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))) = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
16 eqid 2738 . . 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 2738 . . 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 38010 . 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 5033 . . . . . 6 (𝑠 = 𝑡 → (𝑠 𝑊𝑡 𝑊))
2019notbid 321 . . . . 5 (𝑠 = 𝑡 → (¬ 𝑠 𝑊 ↔ ¬ 𝑡 𝑊))
21 oveq1 7177 . . . . . 6 (𝑠 = 𝑡 → (𝑠 (𝑋 𝑊)) = (𝑡 (𝑋 𝑊)))
2221eqeq1d 2740 . . . . 5 (𝑠 = 𝑡 → ((𝑠 (𝑋 𝑊)) = 𝑋 ↔ (𝑡 (𝑋 𝑊)) = 𝑋))
2320, 22anbi12d 634 . . . 4 (𝑠 = 𝑡 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ↔ (¬ 𝑡 𝑊 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)))
2412oveq1i 7180 . . . . 5 (𝐶 (𝑋 𝑊)) = (if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) (𝑋 𝑊))
25 breq1 5033 . . . . . . 7 (𝑠 = 𝑡 → (𝑠 (𝑃 𝑄) ↔ 𝑡 (𝑃 𝑄)))
26 oveq1 7177 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑡 → (𝑠 𝑧) = (𝑡 𝑧))
2726oveq1d 7185 . . . . . . . . . . . . . . 15 (𝑠 = 𝑡 → ((𝑠 𝑧) 𝑊) = ((𝑡 𝑧) 𝑊))
2827oveq2d 7186 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → (𝑍 ((𝑠 𝑧) 𝑊)) = (𝑍 ((𝑡 𝑧) 𝑊)))
2928oveq2d 7186 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊))) = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))
3010, 29syl5eq 2785 . . . . . . . . . . . 12 (𝑠 = 𝑡𝑁 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))
3130eqeq2d 2749 . . . . . . . . . . 11 (𝑠 = 𝑡 → (𝑢 = 𝑁𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))))
3231imbi2d 344 . . . . . . . . . 10 (𝑠 = 𝑡 → (((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))))
3332ralbidv 3109 . . . . . . . . 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 7129 . . . . . . . 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 2785 . . . . . . 7 (𝑠 = 𝑡𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))))
36 oveq1 7177 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑠 𝑈) = (𝑡 𝑈))
37 oveq2 7178 . . . . . . . . . . 11 (𝑠 = 𝑡 → (𝑃 𝑠) = (𝑃 𝑡))
3837oveq1d 7185 . . . . . . . . . 10 (𝑠 = 𝑡 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑡) 𝑊))
3938oveq2d 7186 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑡) 𝑊)))
4036, 39oveq12d 7188 . . . . . . . 8 (𝑠 = 𝑡 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))))
418, 40syl5eq 2785 . . . . . . 7 (𝑠 = 𝑡𝐹 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))))
4225, 35, 41ifbieq12d 4442 . . . . . 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 7185 . . . . 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 2785 . . . 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 5272 . . 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 232 . 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 65 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 399  w3a 1088   = wceq 1542  wcel 2114  wne 2934  wral 3053  wrex 3054  ifcif 4414   class class class wbr 5030  cfv 6339  crio 7126  (class class class)co 7170  Basecbs 16586  lecple 16675  joincjn 17670  meetcmee 17671  Atomscatm 36900  HLchlt 36987  LHypclh 37621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-riotaBAD 36590
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-undef 7968  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-p1 17766  df-lat 17772  df-clat 17834  df-oposet 36813  df-ol 36815  df-oml 36816  df-covers 36903  df-ats 36904  df-atl 36935  df-cvlat 36959  df-hlat 36988  df-llines 37135  df-lplanes 37136  df-lvols 37137  df-lines 37138  df-psubsp 37140  df-pmap 37141  df-padd 37433  df-lhyp 37625
This theorem is referenced by:  cdleme29c  38013
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