Theorem nfiseq 9058

Description: Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)

Hypotheses

Ref	Expression
nfiseq.1	⊢ Ⅎ𝑥𝑀
nfiseq.2	⊢ Ⅎ𝑥 +
nfiseq.3	⊢ Ⅎ𝑥𝐹
nfiseq.4	⊢ Ⅎ𝑥𝑆

Assertion

Ref	Expression
nfiseq	⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆)

Proof of Theorem nfiseq

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

Step	Hyp	Ref	Expression
1		df-iseq 9052	. 2 ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2178	. . . . . 6 ⊢ Ⅎ𝑥ℤ≥
3		nfiseq.1	. . . . . 6 ⊢ Ⅎ𝑥𝑀
4	2, 3	nffv 5146	. . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀)
5		nfiseq.4	. . . . 5 ⊢ Ⅎ𝑥𝑆
6		nfcv 2178	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 + 1)
7		nfcv 2178	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
8		nfiseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
9		nfiseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10	9, 6	nffv 5146	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑦 + 1))
11	7, 8, 10	nfov 5496	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + (𝐹‘(𝑦 + 1)))
12	6, 11	nfop 3561	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩
13	4, 5, 12	nfmpt2 5534	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩)
14	9, 3	nffv 5146	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
15	3, 14	nfop 3561	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 5943	. . 3 ⊢ Ⅎ𝑥frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 4540	. 2 ⊢ Ⅎ𝑥ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2175	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆)

Syntax hints:  Ⅎwnfc 2165  ⟨cop 3375  ran crn 4307  ‘cfv 4863  (class class class)co 5473   ↦ cmpt2 5475  freccfrec 5938  1c1 6833   + caddc 6835  ℤ_≥cuz 8407  seqcseq 9051

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-un 2919  df-in 2921  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-br 3761  df-opab 3815  df-mpt 3816  df-xp 4312  df-cnv 4314  df-dm 4316  df-rn 4317  df-res 4318  df-iota 4828  df-fv 4871  df-ov 5476  df-oprab 5477  df-mpt2 5478  df-recs 5881  df-frec 5939  df-iseq 9052

This theorem is referenced by:  nfsum1  9714  nfsum  9715