Theorem nfiseq 8898

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 8893	. 2 ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((y ∈ (ℤ≥‘𝑀), z ∈ 𝑆 ↦ ⟨(y + 1), (z + (𝐹‘(y + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2175	. . . . . 6 ⊢ Ⅎxℤ≥
3		nfiseq.1	. . . . . 6 ⊢ Ⅎx𝑀
4	2, 3	nffv 5128	. . . . 5 ⊢ Ⅎx(ℤ≥‘𝑀)
5		nfiseq.4	. . . . 5 ⊢ Ⅎx𝑆
6		nfcv 2175	. . . . . 6 ⊢ Ⅎx(y + 1)
7		nfcv 2175	. . . . . . 7 ⊢ Ⅎxz
8		nfiseq.2	. . . . . . 7 ⊢ Ⅎx +
9		nfiseq.3	. . . . . . . 8 ⊢ Ⅎx𝐹
10	9, 6	nffv 5128	. . . . . . 7 ⊢ Ⅎx(𝐹‘(y + 1))
11	7, 8, 10	nfov 5478	. . . . . 6 ⊢ Ⅎx(z + (𝐹‘(y + 1)))
12	6, 11	nfop 3556	. . . . 5 ⊢ Ⅎx⟨(y + 1), (z + (𝐹‘(y + 1)))⟩
13	4, 5, 12	nfmpt2 5515	. . . 4 ⊢ Ⅎx(y ∈ (ℤ≥‘𝑀), z ∈ 𝑆 ↦ ⟨(y + 1), (z + (𝐹‘(y + 1)))⟩)
14	9, 3	nffv 5128	. . . . 5 ⊢ Ⅎx(𝐹‘𝑀)
15	3, 14	nfop 3556	. . . 4 ⊢ Ⅎx⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 5921	. . 3 ⊢ Ⅎxfrec((y ∈ (ℤ≥‘𝑀), z ∈ 𝑆 ↦ ⟨(y + 1), (z + (𝐹‘(y + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 4522	. 2 ⊢ Ⅎxran frec((y ∈ (ℤ≥‘𝑀), z ∈ 𝑆 ↦ ⟨(y + 1), (z + (𝐹‘(y + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2172	1 ⊢ Ⅎxseq𝑀( + , 𝐹, 𝑆)

Syntax hints:  Ⅎwnfc 2162  ⟨cop 3370  ran crn 4289  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  freccfrec 5917  1c1 6712   + caddc 6714  ℤ_≥cuz 8249  seqcseq 8892

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-recs 5861  df-frec 5918  df-iseq 8893

This theorem is referenced by: (None)