Theorem nfseq 13936

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 13927	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1)
4		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
5		nfseq.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
6		nfseq.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
7	6, 3	nffv 6836	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1))
8	4, 5, 7	nfov 7383	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
9	3, 8	nfop 4843	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
10	2, 2, 9	nfmpo 7435	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11		nfseq.1	. . . . 5 ⊢ Ⅎ𝑥𝑀
12	6, 11	nffv 6836	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
13	11, 12	nfop 4843	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
14	10, 13	nfrdg 8343	. . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		nfcv 2891	. . 3 ⊢ Ⅎ𝑥ω
16	14, 15	nfima 6023	. 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
17	1, 16	nfcxfr 2889	1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2876  Vcvv 3438  ⟨cop 4585   “ cima 5626  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  ωcom 7806  reccrdg 8338  1c1 11029   + caddc 11031  seqcseq 13926

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-xp 5629  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-iota 6442  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-seq 13927

This theorem is referenced by:  seqof2  13985  nfsum1  15615  nfsum  15616  nfcprod1  15833  nfcprod  15834  lgamgulm2  26962  binomcxplemdvbinom  44326  binomcxplemdvsum  44328  binomcxplemnotnn0  44329  fmuldfeqlem1  45564  fmuldfeq  45565  sumnnodd  45612  stoweidlem51  46033