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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 2821 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
3 | 1, 2 | esumeq12d 31313 | 1 ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Σ*cesum 31307 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-iota 6307 df-fv 6356 df-ov 7152 df-esum 31308 |
This theorem is referenced by: esumrnmpt 31332 esumpad 31335 esumpad2 31336 esumpr 31346 esumpr2 31347 esumfzf 31349 esumpmono 31359 esumcvg 31366 esumcvg2 31367 esum2dlem 31372 measvun 31489 ddemeas 31516 oms0 31576 omssubadd 31579 carsgsigalem 31594 carsgclctunlem1 31596 carsgclctunlem2 31598 carsgclctun 31600 pmeasmono 31603 pmeasadd 31604 |
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