Theorem nfseq 10069

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 10060	. 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2240	. . . . . 6 ⊢ Ⅎ𝑥ℤ≥
3		nfseq.1	. . . . . 6 ⊢ Ⅎ𝑥𝑀
4	2, 3	nffv 5363	. . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀)
5		nfcv 2240	. . . . 5 ⊢ Ⅎ𝑥V
6		nfcv 2240	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1)
7		nfcv 2240	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
8		nfseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
9		nfseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10	9, 6	nffv 5363	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1))
11	7, 8, 10	nfov 5733	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
12	6, 11	nfop 3668	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
13	4, 5, 12	nfmpo 5772	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
14	9, 3	nffv 5363	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
15	3, 14	nfop 3668	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 6223	. . 3 ⊢ Ⅎ𝑥frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 4722	. 2 ⊢ Ⅎ𝑥ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2237	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2227  Vcvv 2641  ⟨cop 3477  ran crn 4478  ‘cfv 5059  (class class class)co 5706   ∈ cmpo 5708  freccfrec 6217  1c1 7501   + caddc 7503  ℤ_≥cuz 9176  seqcseq 10059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-un 3025  df-in 3027  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-iota 5024  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-recs 6132  df-frec 6218  df-seqfrec 10060

This theorem is referenced by:  seq3f1olemstep  10115  seq3f1olemp  10116  nfsum1  10964  nfsum  10965