Theorem nfesum1 33790

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 33778	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2891	. . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2891	. . . 4 ⊢ Ⅎ𝑘 tsums
4		nfmpt1 5257	. . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵)
5	2, 3, 4	nfov 7449	. . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
6	5	nfuni 4916	. 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
7	1, 6	nfcxfr 2889	1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2875  ∪ cuni 4909   ↦ cmpt 5232  (class class class)co 7419  0cc0 11140  +∞cpnf 11277  [,]cicc 13362   ↾_s cress 17212  ℝ^*_𝑠cxrs 17485   tsums ctsu 24074  Σ^*cesum 33777

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 2696

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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-iota 6501  df-fv 6557  df-ov 7422  df-esum 33778

This theorem is referenced by:  esumfsup  33820  esum2d  33843  oms0  34048  omssubadd  34051