Theorem nfseq 10896

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 10887	. 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2392	. . . . . 6 ⊢ Ⅎ𝑥ℤ≥
3		nfseq.1	. . . . . 6 ⊢ Ⅎ𝑥𝑀
4	2, 3	nffv 5705	. . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀)
5		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑥V
6		nfcv 2392	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1)
7		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
8		nfseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
9		nfseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10	9, 6	nffv 5705	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1))
11	7, 8, 10	nfov 6115	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
12	6, 11	nfop 3920	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
13	4, 5, 12	nfmpo 6157	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
14	9, 3	nffv 5705	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
15	3, 14	nfop 3920	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 6667	. . 3 ⊢ Ⅎ𝑥frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 5027	. 2 ⊢ Ⅎ𝑥ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2389	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

This proof depends on syntax axioms:  Ⅎwnfc 2379  Vcvv 2821  ⟨cop 3712  ran crn 4775  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087  freccfrec 6661  1c1 8180   + caddc 8182  ℤ_≥cuz 9923  seqcseq 10886

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-recs 6576  df-frec 6662  df-seqfrec 10887

This theorem is used by:  seq3f1olemstep  10953  seq3f1olemp  10954  nfsum1  12124  nfsum  12125  nfcprod1  12323  nfcprod  12324