Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme27b Structured version   Visualization version   GIF version

Theorem cdleme27b 35973
Description: Lemma for cdleme27N 35974. (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 4688 . . 3 (𝑠 = 𝑡 → (𝑠 (𝑃 𝑄) ↔ 𝑡 (𝑃 𝑄)))
2 oveq1 6697 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (𝑠 𝑧) = (𝑡 𝑧))
32oveq1d 6705 . . . . . . . . . . 11 (𝑠 = 𝑡 → ((𝑠 𝑧) 𝑊) = ((𝑡 𝑧) 𝑊))
43oveq2d 6706 . . . . . . . . . 10 (𝑠 = 𝑡 → (𝑍 ((𝑠 𝑧) 𝑊)) = (𝑍 ((𝑡 𝑧) 𝑊)))
54oveq2d 6706 . . . . . . . . 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 2710 . . . . . . . 8 (𝑠 = 𝑡𝑁 = 𝑂)
98eqeq2d 2661 . . . . . . 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 3015 . . . . 5 (𝑠 = 𝑡 → (∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁) ↔ ∀𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂)))
1211riotabidv 6653 . . . 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 2710 . . 3 (𝑠 = 𝑡𝐷 = 𝐸)
16 oveq1 6697 . . . . 5 (𝑠 = 𝑡 → (𝑠 𝑈) = (𝑡 𝑈))
17 oveq2 6698 . . . . . . 7 (𝑠 = 𝑡 → (𝑃 𝑠) = (𝑃 𝑡))
1817oveq1d 6705 . . . . . 6 (𝑠 = 𝑡 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑡) 𝑊))
1918oveq2d 6706 . . . . 5 (𝑠 = 𝑡 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑡) 𝑊)))
2016, 19oveq12d 6708 . . . 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 2710 . . 3 (𝑠 = 𝑡𝐹 = 𝐺)
241, 15, 23ifbieq12d 4146 . 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 2710 1 (𝑠 = 𝑡𝐶 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wral 2941  ifcif 4119   class class class wbr 4685  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
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-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-riota 6651  df-ov 6693
This theorem is referenced by:  cdleme27N  35974  cdleme28c  35977
  Copyright terms: Public domain W3C validator