Theorem nfseq 14029

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-seq 14020	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1)
4		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
5		nfseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
6		nfseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
7	6, 3	nffv 6886	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1))
8	4, 5, 7	nfov 7435	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
9	3, 8	nfop 4865	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
10	2, 2, 9	nfmpo 7489	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11		nfseq.1	. . . . 5 ⊢ Ⅎ𝑥𝑀
12	6, 11	nffv 6886	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
13	11, 12	nfop 4865	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
14	10, 13	nfrdg 8428	. . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		nfcv 2898	. . 3 ⊢ Ⅎ𝑥ω
16	14, 15	nfima 6055	. 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
17	1, 16	nfcxfr 2896	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2883  Vcvv 3459  ⟨cop 4607   “ cima 5657  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  ωcom 7861  reccrdg 8423  1c1 11130   + caddc 11132  seqcseq 14019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-xp 5660  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-iota 6484  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seq 14020

This theorem is referenced by:  seqof2  14078  nfsum1  15706  nfsum  15707  nfcprod1  15924  nfcprod  15925  lgamgulm2  26998  binomcxplemdvbinom  44377  binomcxplemdvsum  44379  binomcxplemnotnn0  44380  fmuldfeqlem1  45611  fmuldfeq  45612  sumnnodd  45659  stoweidlem51  46080