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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 1537 Ⅎwnf 1784 ∈ wcel 2114 Σ*cesum 31286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-iota 6314 df-fv 6363 df-ov 7159 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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