Theorem nfiseq 9528

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 9522	. 2 ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2220	. . . . . 6 ⊢ Ⅎ𝑥ℤ≥
3		nfiseq.1	. . . . . 6 ⊢ Ⅎ𝑥𝑀
4	2, 3	nffv 5216	. . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀)
5		nfiseq.4	. . . . 5 ⊢ Ⅎ𝑥𝑆
6		nfcv 2220	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 + 1)
7		nfcv 2220	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
8		nfiseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
9		nfiseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10	9, 6	nffv 5216	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑦 + 1))
11	7, 8, 10	nfov 5566	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + (𝐹‘(𝑦 + 1)))
12	6, 11	nfop 3594	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩
13	4, 5, 12	nfmpt2 5604	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩)
14	9, 3	nffv 5216	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
15	3, 14	nfop 3594	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 6045	. . 3 ⊢ Ⅎ𝑥frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 4607	. 2 ⊢ Ⅎ𝑥ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2217	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆)

Syntax hints:  Ⅎwnfc 2207  ⟨cop 3409  ran crn 4372  ‘cfv 4932  (class class class)co 5543   ↦ cmpt2 5545  freccfrec 6039  1c1 7044   + caddc 7046  ℤ_≥cuz 8700  seqcseq 9521

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 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-recs 5954  df-frec 6040  df-iseq 9522

This theorem is referenced by:  nfsum1  10331  nfsum  10332