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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 2212 | . . . . 5 ⊢ (𝜑 → ∀𝑘 𝐴 = 𝐵) |
4 | esumeq12dvaf.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) | |
5 | 4 | ex 415 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 = 𝐷)) |
6 | 1, 5 | ralrimi 3215 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) |
7 | mpteq12f 5142 | . . . . 5 ⊢ ((∀𝑘 𝐴 = 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷)) | |
8 | 3, 6, 7 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐷)) |
9 | 8 | oveq2d 7165 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) |
10 | 9 | unieqd 4845 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) |
11 | df-esum 31308 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
12 | df-esum 31308 | . 2 ⊢ Σ*𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)) | |
13 | 10, 11, 12 | 3eqtr4g 2880 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1534 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 ∀wral 3137 ∪ cuni 4831 ↦ cmpt 5139 (class class class)co 7149 0cc0 10530 +∞cpnf 10665 [,]cicc 12735 ↾s cress 16479 ℝ*𝑠cxrs 16768 tsums ctsu 22729 Σ*cesum 31307 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-iota 6307 df-fv 6356 df-ov 7152 df-esum 31308 |
This theorem is referenced by: esumeq12dva 31312 esumeq1d 31315 esumeq2d 31317 esumpinfval 31353 measvunilem0 31493 |
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