Theorem nfesum1 34204

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 34192	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		nfcv 2899	. . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞))
3		nfcv 2899	. . . 4 ⊢ Ⅎ𝑘 tsums
4		nfmpt1 5185	. . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵)
5	2, 3, 4	nfov 7392	. . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
6	5	nfuni 4858	. 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
7	1, 6	nfcxfr 2897	1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵

Syntax hints:  Ⅎwnfc 2884  ∪ cuni 4851   ↦ cmpt 5167  (class class class)co 7362  0cc0 11033  +∞cpnf 11171  [,]cicc 13296   ↾_s cress 17195  ℝ^*_𝑠cxrs 17459   tsums ctsu 24105  Σ^*cesum 34191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-iota 6450  df-fv 6502  df-ov 7365  df-esum 34192

This theorem is referenced by:  esumfsup  34234  esum2d  34257  oms0  34461  omssubadd  34464