Theorem nfesum2 30913

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 30900	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2951	. . . 4 ⊢ Ⅎ𝑥(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2951	. . . 4 ⊢ Ⅎ𝑥 tsums
4		nfesum2.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfesum2.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfmpt 5064	. . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵)
7	2, 3, 6	nfov 7053	. . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
8	7	nfuni 4757	. 2 ⊢ Ⅎ𝑥∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
9	1, 8	nfcxfr 2949	1 ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2935  ∪ cuni 4751   ↦ cmpt 5047  (class class class)co 7023  0cc0 10390  +∞cpnf 10525  [,]cicc 12595   ↾_s cress 16317  ℝ^*_𝑠cxrs 16606   tsums ctsu 22421  Σ^*cesum 30899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-iota 6196  df-fv 6240  df-ov 7026  df-esum 30900

This theorem is referenced by:  esum2dlem  30964