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| Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq2d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.) |
| Ref | Expression |
|---|---|
| esumeq2d.0 | ⊢ Ⅎ𝑘𝜑 |
| esumeq2d.1 | ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| esumeq2d | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | esumeq2d.0 | . 2 ⊢ Ⅎ𝑘𝜑 | |
| 2 | eqidd 2738 | . 2 ⊢ (𝜑 → 𝐴 = 𝐴) | |
| 3 | esumeq2d.1 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) | |
| 4 | 3 | r19.21bi 3251 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) |
| 5 | 1, 2, 4 | esumeq12dvaf 34032 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Ⅎwnf 1783 ∀wral 3061 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: esumeq2dv 34039 esumpad 34056 esumlef 34063 esumrnmpt2 34069 voliune 34230 omssubadd 34302 carsggect 34320 omsmeas 34325 dstrvprob 34474 |
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