![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 2206 | . . . . 5 ⊢ (𝜑 → ∀𝑘 𝐴 = 𝐵) |
4 | esumeq12dvaf.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) | |
5 | 4 | ex 413 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 = 𝐷)) |
6 | 1, 5 | ralrimi 3240 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 = 𝐷) |
7 | mpteq12f 5193 | . . . . 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 7372 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) |
10 | 9 | unieqd 4879 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷))) |
11 | df-esum 32567 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
12 | df-esum 32567 | . 2 ⊢ Σ*𝑘 ∈ 𝐵𝐷 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐷)) | |
13 | 10, 11, 12 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ∀wral 3064 ∪ cuni 4865 ↦ cmpt 5188 (class class class)co 7356 0cc0 11050 +∞cpnf 11185 [,]cicc 13266 ↾s cress 17111 ℝ*𝑠cxrs 17381 tsums ctsu 23475 Σ*cesum 32566 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-iota 6448 df-fv 6504 df-ov 7359 df-esum 32567 |
This theorem is referenced by: esumeq12dva 32571 esumeq1d 32574 esumeq2d 32576 esumpinfval 32612 measvunilem0 32752 |
Copyright terms: Public domain | W3C validator |