Users' Mathboxes Mathbox for Gino Giotto < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbvditgvw2 Structured version   Visualization version   GIF version

Theorem cbvditgvw2 36192
Description: Change bound variable and domain in a directed integral, using implicit substitution. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
cbvditgvw2.1 𝐴 = 𝐵
cbvditgvw2.2 𝐶 = 𝐷
cbvditgvw2.3 (𝑥 = 𝑦𝐸 = 𝐹)
Assertion
Ref Expression
cbvditgvw2 ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐹 d𝑦
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷   𝑦,𝐸   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)   𝐸(𝑥)   𝐹(𝑦)

Proof of Theorem cbvditgvw2
StepHypRef Expression
1 cbvditgvw2.1 . . . 4 𝐴 = 𝐵
2 cbvditgvw2.2 . . . 4 𝐶 = 𝐷
31, 2breq12i 5158 . . 3 (𝐴𝐶𝐵𝐷)
4 cbvditgvw2.3 . . . 4 (𝑥 = 𝑦𝐸 = 𝐹)
51, 2oveq12i 7437 . . . . 5 (𝐴(,)𝐶) = (𝐵(,)𝐷)
65a1i 11 . . . 4 (𝑥 = 𝑦 → (𝐴(,)𝐶) = (𝐵(,)𝐷))
74, 6cbvitgvw2 36191 . . 3 ∫(𝐴(,)𝐶)𝐸 d𝑥 = ∫(𝐵(,)𝐷)𝐹 d𝑦
82a1i 11 . . . . . 6 (𝑥 = 𝑦𝐶 = 𝐷)
91a1i 11 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝐵)
108, 9oveq12d 7443 . . . . 5 (𝑥 = 𝑦 → (𝐶(,)𝐴) = (𝐷(,)𝐵))
114, 10cbvitgvw2 36191 . . . 4 ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑦
1211negeqi 11492 . . 3 -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑦
133, 7, 12ifbieq12i 4557 . 2 if(𝐴𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥) = if(𝐵𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
14 df-ditg 25878 . 2 ⨜[𝐴𝐶]𝐸 d𝑥 = if(𝐴𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥)
15 df-ditg 25878 . 2 ⨜[𝐵𝐷]𝐹 d𝑦 = if(𝐵𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
1613, 14, 153eqtr4i 2771 1 ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐹 d𝑦
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1535  ifcif 4530   class class class wbr 5149  (class class class)co 7425  cle 11287  -cneg 11484  (,)cioo 13377  citg 25648  cdit 25877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6317  df-iota 6510  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-neg 11486  df-seq 14029  df-sum 15709  df-itg 25653  df-ditg 25878
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator