![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2913 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2913 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | cbvsumi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | cbvsumi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvsum 14629 | 1 ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 Ⅎwnfc 2900 Σcsu 14620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-xp 5255 df-cnv 5257 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-iota 5992 df-fv 6037 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-seq 13005 df-sum 14621 |
This theorem is referenced by: sumfc 14644 sumss2 14661 fsumzcl2 14673 fsumsplitf 14676 sumsnf 14677 sumsn 14679 sumsns 14683 fsummsnunz 14687 fsumsplitsnun 14688 fsummsnunzOLD 14689 fsumsplitsnunOLD 14690 fsum2dlem 14705 fsumcom2 14709 fsumshftm 14716 fsumrlim 14746 fsumo1 14747 o1fsum 14748 fsumiun 14756 ovolfiniun 23485 ovoliun2 23490 volfiniun 23531 itgfsum 23809 elplyd 24174 coeeq2 24214 fsumdvdscom 25128 fsumdvdsmul 25138 fsumvma 25155 fsumshftd 34756 binomcxplemdvsum 39077 sumsnd 39704 fourierdlem115 40952 fsummsndifre 41867 fsumsplitsndif 41868 fsummmodsndifre 41869 fsummmodsnunz 41870 |
Copyright terms: Public domain | W3C validator |