Theorem nfseq 10494

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-seqfrec 10485	. 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2332	. . . . . 6 ⊢ Ⅎ𝑥ℤ≥
3		nfseq.1	. . . . . 6 ⊢ Ⅎ𝑥𝑀
4	2, 3	nffv 5547	. . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀)
5		nfcv 2332	. . . . 5 ⊢ Ⅎ𝑥V
6		nfcv 2332	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1)
7		nfcv 2332	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
8		nfseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
9		nfseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10	9, 6	nffv 5547	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1))
11	7, 8, 10	nfov 5930	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
12	6, 11	nfop 3812	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
13	4, 5, 12	nfmpo 5969	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
14	9, 3	nffv 5547	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
15	3, 14	nfop 3812	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 6425	. . 3 ⊢ Ⅎ𝑥frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 4893	. 2 ⊢ Ⅎ𝑥ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2329	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2319  Vcvv 2752  ⟨cop 3613  ran crn 4648  ‘cfv 5238  (class class class)co 5900   ∈ cmpo 5902  freccfrec 6419  1c1 7847   + caddc 7849  ℤ_≥cuz 9563  seqcseq 10484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-xp 4653  df-cnv 4655  df-dm 4657  df-rn 4658  df-res 4659  df-iota 5199  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-recs 6334  df-frec 6420  df-seqfrec 10485

This theorem is referenced by:  seq3f1olemstep  10540  seq3f1olemp  10541  nfsum1  11405  nfsum  11406  nfcprod1  11603  nfcprod  11604