Theorem nfseq 13731

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 13722	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1)
4		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
5		nfseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
6		nfseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
7	6, 3	nffv 6784	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1))
8	4, 5, 7	nfov 7305	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
9	3, 8	nfop 4820	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
10	2, 2, 9	nfmpo 7357	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11		nfseq.1	. . . . 5 ⊢ Ⅎ𝑥𝑀
12	6, 11	nffv 6784	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
13	11, 12	nfop 4820	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
14	10, 13	nfrdg 8245	. . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		nfcv 2907	. . 3 ⊢ Ⅎ𝑥ω
16	14, 15	nfima 5977	. 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
17	1, 16	nfcxfr 2905	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2887  Vcvv 3432  ⟨cop 4567   “ cima 5592  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ωcom 7712  reccrdg 8240  1c1 10872   + caddc 10874  seqcseq 13721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-xp 5595  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-iota 6391  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seq 13722

This theorem is referenced by:  seqof2  13781  nfsum1  15401  nfsum  15402  nfsumOLD  15403  nfcprod1  15620  nfcprod  15621  lgamgulm2  26185  binomcxplemdvbinom  41971  binomcxplemdvsum  41973  binomcxplemnotnn0  41974  fmuldfeqlem1  43123  fmuldfeq  43124  sumnnodd  43171  stoweidlem51  43592