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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 15572 | . 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 1541 Σcsu 15569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-xp 5639 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-iota 6448 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-seq 13906 df-sum 15570 |
This theorem is referenced by: sumeq12i 15584 fsump1i 15653 fsum2d 15655 fsumxp 15656 isumnn0nn 15726 arisum 15744 arisum2 15745 geo2sum 15757 bpoly0 15932 bpoly1 15933 bpoly2 15939 bpoly3 15940 bpoly4 15941 efsep 15991 ef4p 15994 rpnnen2lem12 16106 ovolicc2lem4 24882 itg10 25050 dveflem 25341 dvply1 25642 vieta1lem2 25669 aaliou3lem4 25704 dvtaylp 25727 pserdvlem2 25785 advlogexp 26008 log2ublem2 26295 log2ublem3 26296 log2ub 26297 ftalem5 26424 cht1 26512 1sgmprm 26545 lgsquadlem2 26727 axlowdimlem16 27853 finsumvtxdg2ssteplem4 28443 rusgrnumwwlks 28866 signsvf0 33132 signsvf1 33133 repr0 33164 k0004val0 42407 binomcxplemnotnn0 42617 fsumiunss 43787 dvnmul 44155 stoweidlem17 44229 dirkertrigeqlem1 44310 etransclem24 44470 etransclem35 44481 |
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