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Theorem nfesum2 32680
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 32667 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
2 nfcv 2908 . . . 4 𝑥(ℝ*𝑠s (0[,]+∞))
3 nfcv 2908 . . . 4 𝑥 tsums
4 nfesum2.1 . . . . 5 𝑥𝐴
5 nfesum2.2 . . . . 5 𝑥𝐵
64, 5nfmpt 5217 . . . 4 𝑥(𝑘𝐴𝐵)
72, 3, 6nfov 7392 . . 3 𝑥((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
87nfuni 4877 . 2 𝑥 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
91, 8nfcxfr 2906 1 𝑥Σ*𝑘𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wnfc 2888   cuni 4870  cmpt 5193  (class class class)co 7362  0cc0 11058  +∞cpnf 11193  [,]cicc 13274  s cress 17119  *𝑠cxrs 17389   tsums ctsu 23493  Σ*cesum 32666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-iota 6453  df-fv 6509  df-ov 7365  df-esum 32667
This theorem is referenced by:  esum2dlem  32731
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