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| Mirrors > Home > MPE Home > Th. List > sumeq1i | Structured version Visualization version GIF version | ||
| Description: Equality inference for sum. (Contributed by NM, 2-Jan-2006.) |
| Ref | Expression |
|---|---|
| sumeq1i.1 | ⊢ 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| sumeq1i | ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sumeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | sumeq1 15725 | . 2 ⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Σcsu 15722 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seq 14043 df-sum 15723 |
| This theorem is referenced by: sumeq12i 15735 fsump1i 15805 fsum2d 15807 fsumxp 15808 isumnn0nn 15878 arisum 15896 arisum2 15897 geo2sum 15909 bpoly0 16086 bpoly1 16087 bpoly2 16093 bpoly3 16094 bpoly4 16095 efsep 16146 ef4p 16149 rpnnen2lem12 16261 ovolicc2lem4 25555 itg10 25723 dveflem 26017 dvply1 26325 vieta1lem2 26353 aaliou3lem4 26388 dvtaylp 26412 pserdvlem2 26472 advlogexp 26697 log2ublem2 26990 log2ublem3 26991 log2ub 26992 ftalem5 27120 cht1 27208 1sgmprm 27243 lgsquadlem2 27425 axlowdimlem16 28972 finsumvtxdg2ssteplem4 29566 rusgrnumwwlks 29994 signsvf0 34595 signsvf1 34596 repr0 34626 sumeq12si 36204 cbvsumvw2 36247 sumcubes 42347 k0004val0 44167 binomcxplemnotnn0 44375 fsumiunss 45590 dvnmul 45958 stoweidlem17 46032 dirkertrigeqlem1 46113 etransclem24 46273 etransclem35 46284 |
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