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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 15033 | . 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 1528 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-iota 6307 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-seq 13358 df-sum 15031 |
This theorem is referenced by: sumeq12i 15045 fsump1i 15112 fsum2d 15114 fsumxp 15115 isumnn0nn 15185 arisum 15203 arisum2 15204 geo2sum 15217 bpoly0 15392 bpoly1 15393 bpoly2 15399 bpoly3 15400 bpoly4 15401 efsep 15451 ef4p 15454 rpnnen2lem12 15566 ovolicc2lem4 24048 itg10 24216 dveflem 24503 dvply1 24800 vieta1lem2 24827 aaliou3lem4 24862 dvtaylp 24885 pserdvlem2 24943 advlogexp 25165 log2ublem2 25452 log2ublem3 25453 log2ub 25454 ftalem5 25581 cht1 25669 1sgmprm 25702 lgsquadlem2 25884 axlowdimlem16 26670 finsumvtxdg2ssteplem4 27257 rusgrnumwwlks 27680 signsvf0 31749 signsvf1 31750 repr0 31781 k0004val0 40382 binomcxplemnotnn0 40565 fsumiunss 41732 dvnmul 42104 stoweidlem17 42179 dirkertrigeqlem1 42260 etransclem24 42420 etransclem35 42431 |
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