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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 2213 | . . . . 5 ⊢ (𝜑 → ∀𝑘 𝐴 = 𝐵) | 
| 4 | esumeq12dvaf.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) | |
| 5 | 4 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 = 𝐷)) | 
| 6 | 1, 5 | ralrimi 3257 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) | 
| 7 | mpteq12f 5230 | . . . . 5 ⊢ ((∀𝑘 𝐴 = 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷)) | |
| 8 | 3, 6, 7 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷)) | 
| 9 | 8 | oveq2d 7447 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) | 
| 10 | 9 | unieqd 4920 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) | 
| 11 | df-esum 34029 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
| 12 | df-esum 34029 | . 2 ⊢ Σ*𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)) | |
| 13 | 10, 11, 12 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ∪ cuni 4907 ↦ cmpt 5225 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 ↾s cress 17274 ℝ*𝑠cxrs 17545 tsums ctsu 24134 Σ*cesum 34028 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-iota 6514 df-fv 6569 df-ov 7434 df-esum 34029 | 
| This theorem is referenced by: esumeq12dva 34033 esumeq1d 34036 esumeq2d 34038 esumpinfval 34074 measvunilem0 34214 | 
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