Theorem nfseq 14067

Description: Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nfseq.1	⊢ Ⅎ𝑥𝑀
nfseq.2	⊢ Ⅎ𝑥 +
nfseq.3	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
nfseq	⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

Proof of Theorem nfseq

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-seq 14058	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2928	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1)
4		nfcv 2928	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
5		nfseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
6		nfseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
7	6, 3	nffv 6898	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1))
8	4, 5, 7	nfov 7453	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
9	3, 8	nfop 4859	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
10	2, 2, 9	nfmpo 7505	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11		nfseq.1	. . . . 5 ⊢ Ⅎ𝑥𝑀
12	6, 11	nffv 6898	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
13	11, 12	nfop 4859	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
14	10, 13	nfrdg 8410	. . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		nfcv 2928	. . 3 ⊢ Ⅎ𝑥ω
16	14, 15	nfima 6075	. 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
17	1, 16	nfcxfr 2926	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

This proof depends on syntax axioms:  Ⅎwnfc 2913  Vcvv 3458  ⟨cop 4600   “ cima 5669  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  ωcom 7871  reccrdg 8405  1c1 11119   + caddc 11121  seqcseq 14057

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-xp 5672  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-iota 6499  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-seq 14058

This theorem is used by:  seqof2  14116  nfsum1  15767  nfsum  15768  nfcprod1  15988  nfcprod  15989  lgamgulm2  27237  binomcxplemdvbinom  45104  binomcxplemdvsum  45106  binomcxplemnotnn0  45107  fmuldfeqlem1  46339  fmuldfeq  46340  sumnnodd  46387  stoweidlem51  46806