Theorem nfiseq 9438

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 9432	. 2 ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2219	. . . . . 6 ⊢ Ⅎ𝑥ℤ≥
3		nfiseq.1	. . . . . 6 ⊢ Ⅎ𝑥𝑀
4	2, 3	nffv 5205	. . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀)
5		nfiseq.4	. . . . 5 ⊢ Ⅎ𝑥𝑆
6		nfcv 2219	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 + 1)
7		nfcv 2219	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
8		nfiseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
9		nfiseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10	9, 6	nffv 5205	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑦 + 1))
11	7, 8, 10	nfov 5555	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + (𝐹‘(𝑦 + 1)))
12	6, 11	nfop 3586	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩
13	4, 5, 12	nfmpt2 5593	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩)
14	9, 3	nffv 5205	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
15	3, 14	nfop 3586	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 6005	. . 3 ⊢ Ⅎ𝑥frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 4597	. 2 ⊢ Ⅎ𝑥ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2216	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆)

Syntax hints:  Ⅎwnfc 2206  ⟨cop 3401  ran crn 4364  ‘cfv 4922  (class class class)co 5532   ↦ cmpt2 5534  freccfrec 6000  1c1 6982   + caddc 6984  ℤ_≥cuz 8619  seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-recs 5943  df-frec 6001  df-iseq 9432

This theorem is referenced by:  nfsum1  10193  nfsum  10194