Theorem nfseq 10687

Description: Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nfseq.1	⊢ Ⅎ𝑥𝑀
nfseq.2	⊢ Ⅎ𝑥 +
nfseq.3	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
nfseq	⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

Proof of Theorem nfseq

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

Step	Hyp	Ref	Expression
1		df-seqfrec 10678	. 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		nfcv 2372	. . . . . 6 ⊢ Ⅎ𝑥ℤ≥
3		nfseq.1	. . . . . 6 ⊢ Ⅎ𝑥𝑀
4	2, 3	nffv 5639	. . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀)
5		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑥V
6		nfcv 2372	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1)
7		nfcv 2372	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
8		nfseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
9		nfseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10	9, 6	nffv 5639	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1))
11	7, 8, 10	nfov 6037	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
12	6, 11	nfop 3873	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
13	4, 5, 12	nfmpo 6079	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
14	9, 3	nffv 5639	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
15	3, 14	nfop 3873	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
16	13, 15	nffrec 6548	. . 3 ⊢ Ⅎ𝑥frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
17	16	nfrn 4969	. 2 ⊢ Ⅎ𝑥ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
18	1, 17	nfcxfr 2369	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2359  Vcvv 2799  ⟨cop 3669  ran crn 4720  ‘cfv 5318  (class class class)co 6007   ∈ cmpo 6009  freccfrec 6542  1c1 8008   + caddc 8010  ℤ_≥cuz 9730  seqcseq 10677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-recs 6457  df-frec 6543  df-seqfrec 10678

This theorem is referenced by:  seq3f1olemstep  10744  seq3f1olemp  10745  nfsum1  11875  nfsum  11876  nfcprod1  12073  nfcprod  12074