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Theorem cdleme27b 34674
 Description: Lemma for cdleme27N 34675. (Contributed by NM, 3-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)
cdleme27.g 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme27.o 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
cdleme27.e 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
cdleme27.y 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
Assertion
Ref Expression
cdleme27b (𝑠 = 𝑡𝐶 = 𝑌)
Distinct variable groups:   𝑡,𝑠,𝑢,𝑧,𝐴   𝐵,𝑠,𝑡,𝑢,𝑧   𝑢,𝐹   𝑢,𝐺   𝐻,𝑠,𝑡,𝑧   ,𝑠,𝑡,𝑢,𝑧   𝐾,𝑠,𝑡,𝑧   ,𝑠,𝑡,𝑢,𝑧   ,𝑠,𝑡,𝑢,𝑧   𝑡,𝑁,𝑢   𝑂,𝑠,𝑢   𝑃,𝑠,𝑡,𝑢,𝑧   𝑄,𝑠,𝑡,𝑢,𝑧   𝑈,𝑠,𝑡,𝑢,𝑧   𝑊,𝑠,𝑡,𝑢,𝑧
Allowed substitution hints:   𝐶(𝑧,𝑢,𝑡,𝑠)   𝐷(𝑧,𝑢,𝑡,𝑠)   𝐸(𝑧,𝑢,𝑡,𝑠)   𝐹(𝑧,𝑡,𝑠)   𝐺(𝑧,𝑡,𝑠)   𝐻(𝑢)   𝐾(𝑢)   𝑁(𝑧,𝑠)   𝑂(𝑧,𝑡)   𝑌(𝑧,𝑢,𝑡,𝑠)   𝑍(𝑧,𝑢,𝑡,𝑠)

Proof of Theorem cdleme27b
StepHypRef Expression
1 breq1 4586 . . 3 (𝑠 = 𝑡 → (𝑠 (𝑃 𝑄) ↔ 𝑡 (𝑃 𝑄)))
2 oveq1 6556 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (𝑠 𝑧) = (𝑡 𝑧))
32oveq1d 6564 . . . . . . . . . . 11 (𝑠 = 𝑡 → ((𝑠 𝑧) 𝑊) = ((𝑡 𝑧) 𝑊))
43oveq2d 6565 . . . . . . . . . 10 (𝑠 = 𝑡 → (𝑍 ((𝑠 𝑧) 𝑊)) = (𝑍 ((𝑡 𝑧) 𝑊)))
54oveq2d 6565 . . . . . . . . 9 (𝑠 = 𝑡 → ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊))) = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊))))
6 cdleme27.n . . . . . . . . 9 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
7 cdleme27.o . . . . . . . . 9 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
85, 6, 73eqtr4g 2669 . . . . . . . 8 (𝑠 = 𝑡𝑁 = 𝑂)
98eqeq2d 2620 . . . . . . 7 (𝑠 = 𝑡 → (𝑢 = 𝑁𝑢 = 𝑂))
109imbi2d 329 . . . . . 6 (𝑠 = 𝑡 → (((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂)))
1110ralbidv 2969 . . . . 5 (𝑠 = 𝑡 → (∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂)))
1211riotabidv 6513 . . . 4 (𝑠 = 𝑡 → (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁)) = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂)))
13 cdleme27.d . . . 4 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
14 cdleme27.e . . . 4 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
1512, 13, 143eqtr4g 2669 . . 3 (𝑠 = 𝑡𝐷 = 𝐸)
16 oveq1 6556 . . . . 5 (𝑠 = 𝑡 → (𝑠 𝑈) = (𝑡 𝑈))
17 oveq2 6557 . . . . . . 7 (𝑠 = 𝑡 → (𝑃 𝑠) = (𝑃 𝑡))
1817oveq1d 6564 . . . . . 6 (𝑠 = 𝑡 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑡) 𝑊))
1918oveq2d 6565 . . . . 5 (𝑠 = 𝑡 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑡) 𝑊)))
2016, 19oveq12d 6567 . . . 4 (𝑠 = 𝑡 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))))
21 cdleme27.f . . . 4 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
22 cdleme27.g . . . 4 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
2320, 21, 223eqtr4g 2669 . . 3 (𝑠 = 𝑡𝐹 = 𝐺)
241, 15, 23ifbieq12d 4063 . 2 (𝑠 = 𝑡 → if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺))
25 cdleme27.c . 2 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
26 cdleme27.y . 2 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
2724, 25, 263eqtr4g 2669 1 (𝑠 = 𝑡𝐶 = 𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ∀wral 2896  ifcif 4036   class class class wbr 4583  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  LHypclh 34288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-riota 6511  df-ov 6552 This theorem is referenced by:  cdleme27N  34675  cdleme28c  34678
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