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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq1d | Structured version Visualization version GIF version |
Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.) |
Ref | Expression |
---|---|
esumeq1d.0 | ⊢ Ⅎ𝑘𝜑 |
esumeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
esumeq1d | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumeq1d.0 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | esumeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | eqidd 2822 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐶) | |
4 | 1, 2, 3 | esumeq12dvaf 31290 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Σ*cesum 31286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-iota 6313 df-fv 6362 df-ov 7158 df-esum 31287 |
This theorem is referenced by: esummono 31313 esumrnmpt2 31327 esumfzf 31328 hasheuni 31344 esum2dlem 31351 measvuni 31473 ddemeas 31495 omssubadd 31558 carsggect 31576 |
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