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Theorem nfesum2 31408
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 31395 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
2 nfcv 2958 . . . 4 𝑥(ℝ*𝑠s (0[,]+∞))
3 nfcv 2958 . . . 4 𝑥 tsums
4 nfesum2.1 . . . . 5 𝑥𝐴
5 nfesum2.2 . . . . 5 𝑥𝐵
64, 5nfmpt 5130 . . . 4 𝑥(𝑘𝐴𝐵)
72, 3, 6nfov 7169 . . 3 𝑥((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
87nfuni 4810 . 2 𝑥 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
91, 8nfcxfr 2956 1 𝑥Σ*𝑘𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wnfc 2939   cuni 4803  cmpt 5113  (class class class)co 7139  0cc0 10530  +∞cpnf 10665  [,]cicc 12733  s cress 16479  *𝑠cxrs 16768   tsums ctsu 22734  Σ*cesum 31394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-iota 6287  df-fv 6336  df-ov 7142  df-esum 31395
This theorem is referenced by:  esum2dlem  31459
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