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Mirrors > Home > ILE Home > Th. List > sumeq1d | GIF version |
Description: Equality deduction for sum. (Contributed by NM, 1-Nov-2005.) |
Ref | Expression |
---|---|
sumeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sumeq1d | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | sumeq1 11129 | . 2 ⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Σcsu 11127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-seqfrec 10224 df-sumdc 11128 |
This theorem is referenced by: sumeq12dv 11146 sumeq12rdv 11147 fsumf1o 11164 fisumss 11166 fsumcllem 11173 fsum1 11186 fzosump1 11191 fsump1 11194 fsum2d 11209 fisumcom2 11212 fsumshftm 11219 fisumrev2 11220 telfsumo 11240 telfsum 11242 telfsum2 11243 fsumparts 11244 fsumiun 11251 bcxmas 11263 isumsplit 11265 isum1p 11266 arisum 11272 arisum2 11273 geoserap 11281 geolim 11285 geo2sum2 11289 cvgratnnlemseq 11300 cvgratnnlemsumlt 11302 mertenslemub 11308 mertenslemi1 11309 mertenslem2 11310 mertensabs 11311 efcvgfsum 11378 eftlub 11401 effsumlt 11403 eirraplem 11488 cvgcmp2nlemabs 13232 trilpolemeq1 13238 |
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