Theorem nfesum2 33665

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 33652	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2898	. . . 4 ⊢ Ⅎ𝑥(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2898	. . . 4 ⊢ Ⅎ𝑥 tsums
4		nfesum2.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfesum2.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfmpt 5257	. . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵)
7	2, 3, 6	nfov 7454	. . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
8	7	nfuni 4917	. 2 ⊢ Ⅎ𝑥∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
9	1, 8	nfcxfr 2896	1 ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2878  ∪ cuni 4910   ↦ cmpt 5233  (class class class)co 7424  0cc0 11144  +∞cpnf 11281  [,]cicc 13365   ↾_s cress 17214  ℝ^*_𝑠cxrs 17487   tsums ctsu 24048  Σ^*cesum 33651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-iota 6503  df-fv 6559  df-ov 7427  df-esum 33652

This theorem is referenced by:  esum2dlem  33716