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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq12dvaf | Structured version Visualization version GIF version |
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.) |
Ref | Expression |
---|---|
esumeq12dvaf.1 | ⊢ Ⅎ𝑘𝜑 |
esumeq12dvaf.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
esumeq12dvaf.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
esumeq12dvaf | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumeq12dvaf.1 | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
2 | esumeq12dvaf.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 1, 2 | alrimi 2211 | . . . . 5 ⊢ (𝜑 → ∀𝑘 𝐴 = 𝐵) |
4 | esumeq12dvaf.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) | |
5 | 4 | ex 416 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 = 𝐷)) |
6 | 1, 5 | ralrimi 3180 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) |
7 | mpteq12f 5113 | . . . . 5 ⊢ ((∀𝑘 𝐴 = 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷)) | |
8 | 3, 6, 7 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷)) |
9 | 8 | oveq2d 7151 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) |
10 | 9 | unieqd 4814 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) |
11 | df-esum 31397 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
12 | df-esum 31397 | . 2 ⊢ Σ*𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)) | |
13 | 10, 11, 12 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ∪ cuni 4800 ↦ cmpt 5110 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 [,]cicc 12729 ↾s cress 16476 ℝ*𝑠cxrs 16765 tsums ctsu 22731 Σ*cesum 31396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-iota 6283 df-fv 6332 df-ov 7138 df-esum 31397 |
This theorem is referenced by: esumeq12dva 31401 esumeq1d 31404 esumeq2d 31406 esumpinfval 31442 measvunilem0 31582 |
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