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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvditgvw2 | Structured version Visualization version GIF version |
Description: Change bound variable and domain in a directed integral, using implicit substitution. (Contributed by GG, 1-Sep-2025.) |
Ref | Expression |
---|---|
cbvditgvw2.1 | ⊢ 𝐴 = 𝐵 |
cbvditgvw2.2 | ⊢ 𝐶 = 𝐷 |
cbvditgvw2.3 | ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹) |
Ref | Expression |
---|---|
cbvditgvw2 | ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvditgvw2.1 | . . . 4 ⊢ 𝐴 = 𝐵 | |
2 | cbvditgvw2.2 | . . . 4 ⊢ 𝐶 = 𝐷 | |
3 | 1, 2 | breq12i 5150 | . . 3 ⊢ (𝐴 ≤ 𝐶 ↔ 𝐵 ≤ 𝐷) |
4 | cbvditgvw2.3 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹) | |
5 | 1, 2 | oveq12i 7441 | . . . . 5 ⊢ (𝐴(,)𝐶) = (𝐵(,)𝐷) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴(,)𝐶) = (𝐵(,)𝐷)) |
7 | 4, 6 | cbvitgvw2 36227 | . . 3 ⊢ ∫(𝐴(,)𝐶)𝐸 d𝑥 = ∫(𝐵(,)𝐷)𝐹 d𝑦 |
8 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
9 | 1 | a1i 11 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
10 | 8, 9 | oveq12d 7447 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶(,)𝐴) = (𝐷(,)𝐵)) |
11 | 4, 10 | cbvitgvw2 36227 | . . . 4 ⊢ ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑦 |
12 | 11 | negeqi 11497 | . . 3 ⊢ -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑦 |
13 | 3, 7, 12 | ifbieq12i 4551 | . 2 ⊢ if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥) = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦) |
14 | df-ditg 25872 | . 2 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥) | |
15 | df-ditg 25872 | . 2 ⊢ ⨜[𝐵 → 𝐷]𝐹 d𝑦 = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦) | |
16 | 13, 14, 15 | 3eqtr4i 2774 | 1 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ifcif 4524 class class class wbr 5141 (class class class)co 7429 ≤ cle 11292 -cneg 11489 (,)cioo 13383 ∫citg 25643 ⨜cdit 25871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-xp 5689 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-iota 6512 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-neg 11491 df-seq 14039 df-sum 15719 df-itg 25648 df-ditg 25872 |
This theorem is referenced by: (None) |
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