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Theorem nfesum2 34042
Description: Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 2-May-2020.)
Hypotheses
Ref Expression
nfesum2.1 𝑥𝐴
nfesum2.2 𝑥𝐵
Assertion
Ref Expression
nfesum2 𝑥Σ*𝑘𝐴𝐵
Distinct variable group:   𝑥,𝑘
Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐵(𝑥,𝑘)

Proof of Theorem nfesum2
StepHypRef Expression
1 df-esum 34029 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
2 nfcv 2905 . . . 4 𝑥(ℝ*𝑠s (0[,]+∞))
3 nfcv 2905 . . . 4 𝑥 tsums
4 nfesum2.1 . . . . 5 𝑥𝐴
5 nfesum2.2 . . . . 5 𝑥𝐵
64, 5nfmpt 5249 . . . 4 𝑥(𝑘𝐴𝐵)
72, 3, 6nfov 7461 . . 3 𝑥((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
87nfuni 4914 . 2 𝑥 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
91, 8nfcxfr 2903 1 𝑥Σ*𝑘𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wnfc 2890   cuni 4907  cmpt 5225  (class class class)co 7431  0cc0 11155  +∞cpnf 11292  [,]cicc 13390  s cress 17274  *𝑠cxrs 17545   tsums ctsu 24134  Σ*cesum 34028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-iota 6514  df-fv 6569  df-ov 7434  df-esum 34029
This theorem is referenced by:  esum2dlem  34093
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