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| Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq1 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) |
| Ref | Expression |
|---|---|
| esumeq1 | ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
| 2 | eqidd 2738 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
| 3 | 1, 2 | esumeq12d 34034 | 1 ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 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: esumrnmpt 34053 esumpad 34056 esumpad2 34057 esumpr 34067 esumpr2 34068 esumfzf 34070 esumpmono 34080 esumcvg 34087 esumcvg2 34088 esum2dlem 34093 measvun 34210 ddemeas 34237 oms0 34299 omssubadd 34302 carsgsigalem 34317 carsgclctunlem1 34319 carsgclctunlem2 34321 carsgclctun 34323 pmeasmono 34326 pmeasadd 34327 |
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