Theorem nfesum2 34219

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 34206	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥 tsums
4		nfesum2.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfesum2.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfmpt 5198	. . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵)
7	2, 3, 6	nfov 7398	. . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
8	7	nfuni 4872	. 2 ⊢ Ⅎ𝑥∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
9	1, 8	nfcxfr 2897	1 ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2884  ∪ cuni 4865   ↦ cmpt 5181  (class class class)co 7368  0cc0 11038  +∞cpnf 11175  [,]cicc 13276   ↾_s cress 17169  ℝ^*_𝑠cxrs 17433   tsums ctsu 24082  Σ^*cesum 34205

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-iota 6456  df-fv 6508  df-ov 7371  df-esum 34206

This theorem is referenced by:  esum2dlem  34270