Theorem nfesum1 34375

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 34363	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2931	. . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2931	. . . 4 ⊢ Ⅎ𝑘 tsums
4		nfmpt1 5212	. . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵)
5	2, 3, 4	nfov 7441	. . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
6	5	nfuni 4881	. 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
7	1, 6	nfcxfr 2929	1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2916  ∪ cuni 4874   ↦ cmpt 5194  (class class class)co 7411  0cc0 11100  +∞cpnf 11240  [,]cicc 13375   ↾_s cress 17290  ℝ^*_𝑠cxrs 17554   tsums ctsu 24252  Σ^*cesum 34362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-iota 6493  df-fv 6545  df-ov 7414  df-esum 34363

This theorem is referenced by:  esumfsup  34405  esum2d  34428  oms0  34632  omssubadd  34635