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Theorem nfesum1 31527
 Description: Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Hypothesis
Ref Expression
nfesum1.1 𝑘𝐴
Assertion
Ref Expression
nfesum1 𝑘Σ*𝑘𝐴𝐵

Proof of Theorem nfesum1
StepHypRef Expression
1 df-esum 31515 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
2 nfcv 2919 . . . 4 𝑘(ℝ*𝑠s (0[,]+∞))
3 nfcv 2919 . . . 4 𝑘 tsums
4 nfmpt1 5130 . . . 4 𝑘(𝑘𝐴𝐵)
52, 3, 4nfov 7180 . . 3 𝑘((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
65nfuni 4805 . 2 𝑘 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
71, 6nfcxfr 2917 1 𝑘Σ*𝑘𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnfc 2899  ∪ cuni 4798   ↦ cmpt 5112  (class class class)co 7150  0cc0 10575  +∞cpnf 10710  [,]cicc 12782   ↾s cress 16542  ℝ*𝑠cxrs 16831   tsums ctsu 22826  Σ*cesum 31514 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 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 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-iota 6294  df-fv 6343  df-ov 7153  df-esum 31515 This theorem is referenced by:  esumfsup  31557  esum2d  31580  oms0  31783  omssubadd  31786
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