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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 31713 | 1 ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 Σ*cesum 31707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-iota 6338 df-fv 6388 df-ov 7216 df-esum 31708 |
This theorem is referenced by: esumrnmpt 31732 esumpad 31735 esumpad2 31736 esumpr 31746 esumpr2 31747 esumfzf 31749 esumpmono 31759 esumcvg 31766 esumcvg2 31767 esum2dlem 31772 measvun 31889 ddemeas 31916 oms0 31976 omssubadd 31979 carsgsigalem 31994 carsgclctunlem1 31996 carsgclctunlem2 31998 carsgclctun 32000 pmeasmono 32003 pmeasadd 32004 |
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