Theorem nfiseq 9747

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 9741	. 2 ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2223	. . . . . 6 ⊢ Ⅎ𝑥ℤ≥
3		nfiseq.1	. . . . . 6 ⊢ Ⅎ𝑥𝑀
4	2, 3	nffv 5260	. . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀)
5		nfiseq.4	. . . . 5 ⊢ Ⅎ𝑥𝑆
6		nfcv 2223	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 + 1)
7		nfcv 2223	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
8		nfiseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
9		nfiseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10	9, 6	nffv 5260	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑦 + 1))
11	7, 8, 10	nfov 5614	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + (𝐹‘(𝑦 + 1)))
12	6, 11	nfop 3612	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩
13	4, 5, 12	nfmpt2 5652	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩)
14	9, 3	nffv 5260	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
15	3, 14	nfop 3612	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 6093	. . 3 ⊢ Ⅎ𝑥frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 4638	. 2 ⊢ Ⅎ𝑥ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2220	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆)

Syntax hints:  Ⅎwnfc 2210  ⟨cop 3425  ran crn 4402  ‘cfv 4969  (class class class)co 5591   ↦ cmpt2 5593  freccfrec 6087  1c1 7254   + caddc 7256  ℤ_≥cuz 8914  seqcseq 9740

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-un 2988  df-in 2990  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-xp 4407  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-iota 4934  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-recs 6002  df-frec 6088  df-iseq 9741

This theorem is referenced by:  nfsum1  10567  nfsum  10568