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Theorem cdleme31sde 39560
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 3901 . . . 4 𝑆 / 𝑡𝐸 = 𝑆 / 𝑡((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
3 nfcvd 2903 . . . . 5 (𝑆𝐴𝑡((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
4 oveq1 7419 . . . . . . . . 9 (𝑡 = 𝑆 → (𝑡 𝑈) = (𝑆 𝑈))
5 oveq2 7420 . . . . . . . . . . 11 (𝑡 = 𝑆 → (𝑃 𝑡) = (𝑃 𝑆))
65oveq1d 7427 . . . . . . . . . 10 (𝑡 = 𝑆 → ((𝑃 𝑡) 𝑊) = ((𝑃 𝑆) 𝑊))
76oveq2d 7428 . . . . . . . . 9 (𝑡 = 𝑆 → (𝑄 ((𝑃 𝑡) 𝑊)) = (𝑄 ((𝑃 𝑆) 𝑊)))
84, 7oveq12d 7430 . . . . . . . 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 2796 . . . . . . 7 (𝑡 = 𝑆𝐷 = 𝑌)
12 oveq2 7420 . . . . . . . 8 (𝑡 = 𝑆 → (𝑠 𝑡) = (𝑠 𝑆))
1312oveq1d 7427 . . . . . . 7 (𝑡 = 𝑆 → ((𝑠 𝑡) 𝑊) = ((𝑠 𝑆) 𝑊))
1411, 13oveq12d 7430 . . . . . 6 (𝑡 = 𝑆 → (𝐷 ((𝑠 𝑡) 𝑊)) = (𝑌 ((𝑠 𝑆) 𝑊)))
1514oveq2d 7428 . . . . 5 (𝑡 = 𝑆 → ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
163, 15csbiegf 3927 . . . 4 (𝑆𝐴𝑆 / 𝑡((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
172, 16eqtrid 2783 . . 3 (𝑆𝐴𝑆 / 𝑡𝐸 = ((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
1817csbeq2dv 3900 . 2 (𝑆𝐴𝑅 / 𝑠𝑆 / 𝑡𝐸 = 𝑅 / 𝑠((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))))
19 eqid 2731 . . 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 39557 . 2 (𝑅𝐴𝑅 / 𝑠((𝑃 𝑄) (𝑌 ((𝑠 𝑆) 𝑊))) = 𝑍)
2218, 21sylan9eqr 2793 1 ((𝑅𝐴𝑆𝐴) → 𝑅 / 𝑠𝑆 / 𝑡𝐸 = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  csb 3893  (class class class)co 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415
This theorem is referenced by:  cdlemefs44  39601  cdlemefs45ee  39605  cdleme17d2  39670
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