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Theorem nfesum2 34022
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 34009 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
2 nfcv 2903 . . . 4 𝑥(ℝ*𝑠s (0[,]+∞))
3 nfcv 2903 . . . 4 𝑥 tsums
4 nfesum2.1 . . . . 5 𝑥𝐴
5 nfesum2.2 . . . . 5 𝑥𝐵
64, 5nfmpt 5255 . . . 4 𝑥(𝑘𝐴𝐵)
72, 3, 6nfov 7461 . . 3 𝑥((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
87nfuni 4919 . 2 𝑥 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
91, 8nfcxfr 2901 1 𝑥Σ*𝑘𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wnfc 2888   cuni 4912  cmpt 5231  (class class class)co 7431  0cc0 11153  +∞cpnf 11290  [,]cicc 13387  s cress 17274  *𝑠cxrs 17547   tsums ctsu 24150  Σ*cesum 34008
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6516  df-fv 6571  df-ov 7434  df-esum 34009
This theorem is referenced by:  esum2dlem  34073
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