Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31sde Structured version   Visualization version   GIF version

Theorem cdleme31sde 34487
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
Hypotheses
Ref Expression
cdleme31sde.c 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme31sde.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdleme31sde.x 𝑌 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme31sde.z 𝑍 = ((𝑃 𝑄) (𝑌 ((𝑅 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme31sde ((𝑅𝐴𝑆𝐴) → 𝑅 / 𝑠𝑆 / 𝑡𝐸 = 𝑍)
Distinct variable groups:   𝑡,𝑠,𝐴   ,𝑠,𝑡   ,𝑠,𝑡   𝑃,𝑠,𝑡   𝑄,𝑠,𝑡   𝑅,𝑠   𝑆,𝑠,𝑡   𝑊,𝑠,𝑡   𝑌,𝑠,𝑡
Allowed substitution hints:   𝐷(𝑡,𝑠)   𝑅(𝑡)   𝑈(𝑡,𝑠)   𝐸(𝑡,𝑠)   𝑍(𝑡,𝑠)

Proof of Theorem cdleme31sde
StepHypRef Expression
1 cdleme31sde.e . . . . 5 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
21csbeq2i 3944 . . . 4 𝑆 / 𝑡𝐸 = 𝑆 / 𝑡((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
3 nfcvd 2751 . . . . 5 (𝑆𝐴𝑡((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
4 oveq1 6534 . . . . . . . . 9 (𝑡 = 𝑆 → (𝑡 𝑈) = (𝑆 𝑈))
5 oveq2 6535 . . . . . . . . . . 11 (𝑡 = 𝑆 → (𝑃 𝑡) = (𝑃 𝑆))
65oveq1d 6542 . . . . . . . . . 10 (𝑡 = 𝑆 → ((𝑃 𝑡) 𝑊) = ((𝑃 𝑆) 𝑊))
76oveq2d 6543 . . . . . . . . 9 (𝑡 = 𝑆 → (𝑄 ((𝑃 𝑡) 𝑊)) = (𝑄 ((𝑃 𝑆) 𝑊)))
84, 7oveq12d 6545 . . . . . . . 8 (𝑡 = 𝑆 → ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))) = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))))
9 cdleme31sde.c . . . . . . . 8 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
10 cdleme31sde.x . . . . . . . 8 𝑌 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
118, 9, 103eqtr4g 2668 . . . . . . 7 (𝑡 = 𝑆𝐷 = 𝑌)
12 oveq2 6535 . . . . . . . 8 (𝑡 = 𝑆 → (𝑠 𝑡) = (𝑠 𝑆))
1312oveq1d 6542 . . . . . . 7 (𝑡 = 𝑆 → ((𝑠 𝑡) 𝑊) = ((𝑠 𝑆) 𝑊))
1411, 13oveq12d 6545 . . . . . 6 (𝑡 = 𝑆 → (𝐷 ((𝑠 𝑡) 𝑊)) = (𝑌 ((𝑠 𝑆) 𝑊)))
1514oveq2d 6543 . . . . 5 (𝑡 = 𝑆 → ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
163, 15csbiegf 3522 . . . 4 (𝑆𝐴𝑆 / 𝑡((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
172, 16syl5eq 2655 . . 3 (𝑆𝐴𝑆 / 𝑡𝐸 = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
1817csbeq2dv 3943 . 2 (𝑆𝐴𝑅 / 𝑠𝑆 / 𝑡𝐸 = 𝑅 / 𝑠((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
19 eqid 2609 . . 3 ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))) = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊)))
20 cdleme31sde.z . . 3 𝑍 = ((𝑃 𝑄) (𝑌 ((𝑅 𝑆) 𝑊)))
2119, 20cdleme31se 34484 . 2 (𝑅𝐴𝑅 / 𝑠((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))) = 𝑍)
2218, 21sylan9eqr 2665 1 ((𝑅𝐴𝑆𝐴) → 𝑅 / 𝑠𝑆 / 𝑡𝐸 = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  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
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-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  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:  cdlemefs44  34528  cdlemefs45ee  34532  cdleme17d2  34597
  Copyright terms: Public domain W3C validator