Theorem nfesum1 34041

Description: Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)

Hypothesis

Ref	Expression
nfesum1.1	⊢ Ⅎ𝑘𝐴

Assertion

Ref	Expression
nfesum1	⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵

Proof of Theorem nfesum1

Step	Hyp	Ref	Expression
1		df-esum 34029	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2905	. . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2905	. . . 4 ⊢ Ⅎ𝑘 tsums
4		nfmpt1 5250	. . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵)
5	2, 3, 4	nfov 7461	. . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
6	5	nfuni 4914	. 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
7	1, 6	nfcxfr 2903	1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2890  ∪ cuni 4907   ↦ cmpt 5225  (class class class)co 7431  0cc0 11155  +∞cpnf 11292  [,]cicc 13390   ↾_s cress 17274  ℝ^*_𝑠cxrs 17545   tsums ctsu 24134  Σ^*cesum 34028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-iota 6514  df-fv 6569  df-ov 7434  df-esum 34029

This theorem is referenced by:  esumfsup  34071  esum2d  34094  oms0  34299  omssubadd  34302