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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 2977 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2977 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | cbvsumi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | cbvsumi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvsum 15046 | 1 ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Ⅎwnfc 2961 Σcsu 15036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-iota 6309 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-seq 13364 df-sum 15037 |
This theorem is referenced by: sumfc 15060 sumss2 15077 fsumzcl2 15089 fsumsplitf 15092 sumsnf 15093 sumsns 15099 fsummsnunz 15103 fsumsplitsnun 15104 fsum2dlem 15119 fsumcom2 15123 fsumshftm 15130 fsumrlim 15160 fsumo1 15161 o1fsum 15162 fsumiun 15170 ovolfiniun 24096 ovoliun2 24101 volfiniun 24142 itgfsum 24421 elplyd 24786 coeeq2 24826 fsumdvdscom 25756 fsumdvdsmul 25766 fsumvma 25783 fsumshftd 36082 binomcxplemdvsum 40680 sumsnd 41276 fourierdlem115 42499 fsummsndifre 43525 fsumsplitsndif 43526 fsummmodsndifre 43527 fsummmodsnunz 43528 |
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