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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 5256 | . . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶) |
7 | 6 | oveq2i 7422 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) |
8 | 7 | unieqi 4920 | . 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) |
9 | df-esum 33324 | . 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) | |
10 | df-esum 33324 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
11 | 8, 9, 10 | 3eqtr4i 2768 | 1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 Ⅎwnfc 2881 ∪ cuni 4907 ↦ cmpt 5230 (class class class)co 7411 0cc0 11112 +∞cpnf 11249 [,]cicc 13331 ↾s cress 17177 ℝ*𝑠cxrs 17450 tsums ctsu 23850 Σ*cesum 33323 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-iota 6494 df-fv 6550 df-ov 7414 df-esum 33324 |
This theorem is referenced by: cbvesumv 33339 esumfzf 33365 carsggect 33615 |
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