Theorem nfesum2 30103

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 30090	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥 tsums
4		nfesum2.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfesum2.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfmpt 4746	. . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵)
7	2, 3, 6	nfov 6676	. . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
8	7	nfuni 4442	. 2 ⊢ Ⅎ𝑥∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
9	1, 8	nfcxfr 2762	1 ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2751  ∪ cuni 4436   ↦ cmpt 4729  (class class class)co 6650  0cc0 9936  +∞cpnf 10071  [,]cicc 12178   ↾_s cress 15858  ℝ^*_𝑠cxrs 16160   tsums ctsu 21929  Σ^*cesum 30089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-ov 6653  df-esum 30090

This theorem is referenced by:  esum2dlem  30154