![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 2726 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
3 | 1, 2 | esumeq12d 33783 | 1 ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Σ*cesum 33777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-iota 6501 df-fv 6557 df-ov 7422 df-esum 33778 |
This theorem is referenced by: esumrnmpt 33802 esumpad 33805 esumpad2 33806 esumpr 33816 esumpr2 33817 esumfzf 33819 esumpmono 33829 esumcvg 33836 esumcvg2 33837 esum2dlem 33842 measvun 33959 ddemeas 33986 oms0 34048 omssubadd 34051 carsgsigalem 34066 carsgclctunlem1 34068 carsgclctunlem2 34070 carsgclctun 34072 pmeasmono 34075 pmeasadd 34076 |
Copyright terms: Public domain | W3C validator |