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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvesum | Structured version Visualization version GIF version |
Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.) |
Ref | Expression |
---|---|
cbvesum.1 | ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) |
cbvesum.2 | ⊢ Ⅎ𝑘𝐴 |
cbvesum.3 | ⊢ Ⅎ𝑗𝐴 |
cbvesum.4 | ⊢ Ⅎ𝑘𝐵 |
cbvesum.5 | ⊢ Ⅎ𝑗𝐶 |
Ref | Expression |
---|---|
cbvesum | ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvesum.3 | . . . . 5 ⊢ Ⅎ𝑗𝐴 | |
2 | cbvesum.2 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
3 | cbvesum.4 | . . . . 5 ⊢ Ⅎ𝑘𝐵 | |
4 | cbvesum.5 | . . . . 5 ⊢ Ⅎ𝑗𝐶 | |
5 | cbvesum.1 | . . . . 5 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) | |
6 | 1, 2, 3, 4, 5 | cbvmptf 5258 | . . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶) |
7 | 6 | oveq2i 7420 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) |
8 | 7 | unieqi 4922 | . 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) |
9 | df-esum 33026 | . 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) | |
10 | df-esum 33026 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
11 | 8, 9, 10 | 3eqtr4i 2771 | 1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Ⅎwnfc 2884 ∪ cuni 4909 ↦ cmpt 5232 (class class class)co 7409 0cc0 11110 +∞cpnf 11245 [,]cicc 13327 ↾s cress 17173 ℝ*𝑠cxrs 17446 tsums ctsu 23630 Σ*cesum 33025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-iota 6496 df-fv 6552 df-ov 7412 df-esum 33026 |
This theorem is referenced by: cbvesumv 33041 esumfzf 33067 carsggect 33317 |
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