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Theorem cdleme31sde 34691
 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 3945 . . . 4 𝑆 / 𝑡𝐸 = 𝑆 / 𝑡((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
3 nfcvd 2752 . . . . 5 (𝑆𝐴𝑡((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
4 oveq1 6556 . . . . . . . . 9 (𝑡 = 𝑆 → (𝑡 𝑈) = (𝑆 𝑈))
5 oveq2 6557 . . . . . . . . . . 11 (𝑡 = 𝑆 → (𝑃 𝑡) = (𝑃 𝑆))
65oveq1d 6564 . . . . . . . . . 10 (𝑡 = 𝑆 → ((𝑃 𝑡) 𝑊) = ((𝑃 𝑆) 𝑊))
76oveq2d 6565 . . . . . . . . 9 (𝑡 = 𝑆 → (𝑄 ((𝑃 𝑡) 𝑊)) = (𝑄 ((𝑃 𝑆) 𝑊)))
84, 7oveq12d 6567 . . . . . . . 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 2669 . . . . . . 7 (𝑡 = 𝑆𝐷 = 𝑌)
12 oveq2 6557 . . . . . . . 8 (𝑡 = 𝑆 → (𝑠 𝑡) = (𝑠 𝑆))
1312oveq1d 6564 . . . . . . 7 (𝑡 = 𝑆 → ((𝑠 𝑡) 𝑊) = ((𝑠 𝑆) 𝑊))
1411, 13oveq12d 6567 . . . . . 6 (𝑡 = 𝑆 → (𝐷 ((𝑠 𝑡) 𝑊)) = (𝑌 ((𝑠 𝑆) 𝑊)))
1514oveq2d 6565 . . . . 5 (𝑡 = 𝑆 → ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
163, 15csbiegf 3523 . . . 4 (𝑆𝐴𝑆 / 𝑡((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
172, 16syl5eq 2656 . . 3 (𝑆𝐴𝑆 / 𝑡𝐸 = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
1817csbeq2dv 3944 . 2 (𝑆𝐴𝑅 / 𝑠𝑆 / 𝑡𝐸 = 𝑅 / 𝑠((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
19 eqid 2610 . . 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 34688 . 2 (𝑅𝐴𝑅 / 𝑠((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))) = 𝑍)
2218, 21sylan9eqr 2666 1 ((𝑅𝐴𝑆𝐴) → 𝑅 / 𝑠𝑆 / 𝑡𝐸 = 𝑍)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⦋csb 3499  (class class class)co 6549 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-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-ov 6552 This theorem is referenced by:  cdlemefs44  34732  cdlemefs45ee  34736  cdleme17d2  34801
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