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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 2764 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2764 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | cbvsumi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | cbvsumi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvsum 14425 | 1 ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 Ⅎwnfc 2751 Σcsu 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 df-sum 14417 |
This theorem is referenced by: sumfc 14440 sumss2 14457 fsumzcl2 14469 fsumsplitf 14472 sumsnf 14473 sumsn 14475 sumsns 14479 fsummsnunz 14483 fsumsplitsnun 14484 fsummsnunzOLD 14485 fsumsplitsnunOLD 14486 fsum2dlem 14501 fsumcom2 14505 fsumcom2OLD 14506 fsumshftm 14513 fsumrlim 14543 fsumo1 14544 o1fsum 14545 fsumiun 14553 ovolfiniun 23269 ovoliun2 23274 volfiniun 23315 itgfsum 23593 elplyd 23958 coeeq2 23998 fsumdvdscom 24911 fsumdvdsmul 24921 fsumvma 24938 fsumshftd 34237 fsumshftdOLD 34238 binomcxplemdvsum 38554 sumsnd 39185 fourierdlem115 40438 fsummsndifre 41342 fsumsplitsndif 41343 fsummmodsndifre 41344 fsummmodsnunz 41345 |
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