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Mirrors > Home > MPE Home > Th. List > cbvsumi | Structured version Visualization version 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 2751 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2751 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | cbvsumi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | cbvsumi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvsum 14273 | 1 ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 Ⅎwnfc 2738 Σcsu 14264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-sum 14265 |
This theorem is referenced by: sumfc 14287 sumss2 14304 fsumzcl2 14316 sumsn 14319 sumsns 14323 fsummsnunz 14327 fsumsplitsnun 14328 fsum2dlem 14343 fsumcom2 14347 fsumcom2OLD 14348 fsumshftm 14355 fsumrlim 14384 fsumo1 14385 o1fsum 14386 fsumiun 14394 ovolfiniun 23076 ovoliun2 23081 volfiniun 23122 itgfsum 23399 elplyd 23762 coeeq2 23802 fsumdvdscom 24711 fsumdvdsmul 24721 fsumvma 24738 fsumshftd 33255 fsumshftdOLD 33256 binomcxplemdvsum 37576 sumsnd 38208 fsumsplitf 38634 sumsnf 38636 fourierdlem115 39114 fsummsndifre 40371 fsumsplitsndif 40372 fsummmodsndifre 40373 fsummmodsnunz 40374 |
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