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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 5156 | . . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶) |
7 | 6 | oveq2i 7156 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) |
8 | 7 | unieqi 4839 | . 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) |
9 | df-esum 31186 | . 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) | |
10 | df-esum 31186 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
11 | 8, 9, 10 | 3eqtr4i 2851 | 1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 Ⅎwnfc 2958 ∪ cuni 4830 ↦ cmpt 5137 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 [,]cicc 12729 ↾s cress 16472 ℝ*𝑠cxrs 16761 tsums ctsu 22661 Σ*cesum 31185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-iota 6307 df-fv 6356 df-ov 7148 df-esum 31186 |
This theorem is referenced by: cbvesumv 31201 esumfzf 31227 carsggect 31475 |
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