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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 11156 | . 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 1332 Σcsu 11154 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-if 3480 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-recs 6210 df-frec 6296 df-seqfrec 10250 df-sumdc 11155 |
This theorem is referenced by: sumeq12dv 11173 sumeq12rdv 11174 fsumf1o 11191 fisumss 11193 fsumcllem 11200 fsum1 11213 fzosump1 11218 fsump1 11221 fsum2d 11236 fisumcom2 11239 fsumshftm 11246 fisumrev2 11247 telfsumo 11267 telfsum 11269 telfsum2 11270 fsumparts 11271 fsumiun 11278 bcxmas 11290 isumsplit 11292 isum1p 11293 arisum 11299 arisum2 11300 geoserap 11308 geolim 11312 geo2sum2 11316 cvgratnnlemseq 11327 cvgratnnlemsumlt 11329 mertenslemub 11335 mertenslemi1 11336 mertenslem2 11337 mertensabs 11338 efcvgfsum 11410 eftlub 11433 effsumlt 11435 eirraplem 11519 cvgcmp2nlemabs 13402 trilpolemeq1 13408 |
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