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Mirrors > Home > ILE Home > Th. List > cbvsumi | GIF version |
Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) |
Ref | Expression |
---|---|
cbvsumi.1 | ⊢ Ⅎ𝑘𝐵 |
cbvsumi.2 | ⊢ Ⅎ𝑗𝐶 |
cbvsumi.3 | ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvsumi | ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvsumi.3 | . 2 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) | |
2 | nfcv 2299 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2299 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | cbvsumi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | cbvsumi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvsum 11268 | 1 ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 Ⅎwnfc 2286 Σcsu 11261 |
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-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-if 3507 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-mpt 4029 df-cnv 4596 df-dm 4598 df-rn 4599 df-res 4600 df-iota 5137 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-recs 6254 df-frec 6340 df-seqfrec 10354 df-sumdc 11262 |
This theorem is referenced by: sumfct 11282 isumss2 11301 fsumzcl2 11313 fsumsplitf 11316 sumsnf 11317 sumsns 11323 fsumsplitsnun 11327 fsum2dlemstep 11342 fisumcom2 11346 fsumshftm 11353 fsumiun 11385 |
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