Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31snd Structured version   Visualization version   GIF version

Theorem cdleme31snd 34488
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Apr-2013.)
Hypotheses
Ref Expression
cdleme31snd.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme31snd.n 𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))
cdleme31snd.e 𝐸 = ((𝑂 𝑈) (𝑄 ((𝑃 𝑂) 𝑊)))
cdleme31snd.o 𝑂 = ((𝑆 𝑉) (𝑃 ((𝑄 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme31snd (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
Distinct variable groups:   𝑣,𝐴   𝑣,𝐷   𝑣,𝑡,   𝑡, ,𝑣   𝑡,𝑂   𝑡,𝑃,𝑣   𝑡,𝑄,𝑣   𝑣,𝑆   𝑡,𝑈,𝑣   𝑣,𝑉   𝑡,𝑊,𝑣
Allowed substitution hints:   𝐴(𝑡)   𝐷(𝑡)   𝑆(𝑡)   𝐸(𝑣,𝑡)   𝑁(𝑣,𝑡)   𝑂(𝑣)   𝑉(𝑡)

Proof of Theorem cdleme31snd
StepHypRef Expression
1 csbnestg 3949 . 2 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝑆 / 𝑣𝑁 / 𝑡𝐷)
2 cdleme31snd.n . . . . 5 𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))
3 cdleme31snd.o . . . . 5 𝑂 = ((𝑆 𝑉) (𝑃 ((𝑄 𝑆) 𝑊)))
42, 3cdleme31sc 34486 . . . 4 (𝑆𝐴𝑆 / 𝑣𝑁 = 𝑂)
54csbeq1d 3505 . . 3 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝑂 / 𝑡𝐷)
6 ovex 6555 . . . . 5 ((𝑆 𝑉) (𝑃 ((𝑄 𝑆) 𝑊))) ∈ V
73, 6eqeltri 2683 . . . 4 𝑂 ∈ V
8 cdleme31snd.d . . . . 5 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
9 cdleme31snd.e . . . . 5 𝐸 = ((𝑂 𝑈) (𝑄 ((𝑃 𝑂) 𝑊)))
108, 9cdleme31sc 34486 . . . 4 (𝑂 ∈ V → 𝑂 / 𝑡𝐷 = 𝐸)
117, 10ax-mp 5 . . 3 𝑂 / 𝑡𝐷 = 𝐸
125, 11syl6eq 2659 . 2 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
131, 12eqtrd 2643 1 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  csb 3498  (class class class)co 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530
This theorem is referenced by:  cdlemeg46ngfr  34620
  Copyright terms: Public domain W3C validator