Theorem nfesum2 31909

Description: Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 2-May-2020.)

Hypotheses

Ref	Expression
nfesum2.1	⊢ Ⅎ𝑥𝐴
nfesum2.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfesum2	⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵

Distinct variable group:   𝑥,𝑘

Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐵(𝑥,𝑘)

Proof of Theorem nfesum2

Step	Hyp	Ref	Expression
1		df-esum 31896	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2906	. . . 4 ⊢ Ⅎ𝑥(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2906	. . . 4 ⊢ Ⅎ𝑥 tsums
4		nfesum2.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfesum2.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfmpt 5177	. . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵)
7	2, 3, 6	nfov 7285	. . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
8	7	nfuni 4843	. 2 ⊢ Ⅎ𝑥∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
9	1, 8	nfcxfr 2904	1 ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2886  ∪ cuni 4836   ↦ cmpt 5153  (class class class)co 7255  0cc0 10802  +∞cpnf 10937  [,]cicc 13011   ↾_s cress 16867  ℝ^*_𝑠cxrs 17128   tsums ctsu 23185  Σ^*cesum 31895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-iota 6376  df-fv 6426  df-ov 7258  df-esum 31896

This theorem is referenced by:  esum2dlem  31960