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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 4676 | . . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐶) |
7 | 6 | oveq2i 6560 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) |
8 | 7 | unieqi 4381 | . 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) |
9 | df-esum 29417 | . 2 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑗 ∈ 𝐴 ↦ 𝐵)) | |
10 | df-esum 29417 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
11 | 8, 9, 10 | 3eqtr4i 2642 | 1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 Ⅎwnfc 2738 ∪ cuni 4372 ↦ cmpt 4643 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 [,]cicc 12049 ↾s cress 15696 ℝ*𝑠cxrs 15983 tsums ctsu 21739 Σ*cesum 29416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-iota 5768 df-fv 5812 df-ov 6552 df-esum 29417 |
This theorem is referenced by: cbvesumv 29432 esumfzf 29458 carsggect 29707 |
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