Theorem nfseq 10709

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	|- F/_ x M
nfseq.2	|- F/_ x .+
nfseq.3	|- F/_ x F

Assertion

Ref	Expression
nfseq	|- F/_ x seq M ( .+ , 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 10700	. 2 |- seq M ( .+ , F ) = ran frec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
2		nfcv 2372	. . . . . 6 |- F/_ x ZZ>=
3		nfseq.1	. . . . . 6 |- F/_ x M
4	2, 3	nffv 5645	. . . . 5 |- F/_ x ( ZZ>= ` M )
5		nfcv 2372	. . . . 5 |- F/_ x _V
6		nfcv 2372	. . . . . 6 |- F/_ x ( z + 1 )
7		nfcv 2372	. . . . . . 7 |- F/_ x w
8		nfseq.2	. . . . . . 7 |- F/_ x .+
9		nfseq.3	. . . . . . . 8 |- F/_ x F
10	9, 6	nffv 5645	. . . . . . 7 |- F/_ x ( F ` ( z + 1 ) )
11	7, 8, 10	nfov 6043	. . . . . 6 |- F/_ x ( w .+ ( F ` ( z + 1 ) ) )
12	6, 11	nfop 3876	. . . . 5 |- F/_ x <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >.
13	4, 5, 12	nfmpo 6085	. . . 4 |- F/_ x ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. )
14	9, 3	nffv 5645	. . . . 5 |- F/_ x ( F ` M )
15	3, 14	nfop 3876	. . . 4 |- F/_ x <. M , ( F ` M ) >.
16	13, 15	nffrec 6557	. . 3 |- F/_ xfrec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
17	16	nfrn 4975	. 2 |- F/_ x ran frec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
18	1, 17	nfcxfr 2369	1 |- F/_ x seq M ( .+ , F )

Syntax hints:   F/_wnfc 2359   _Vcvv 2800   <.cop 3670   ran crn 4724   ` cfv 5324  (class class class)co 6013   e. cmpo 6015  freccfrec 6551   1c1 8023   + caddc 8025   ZZ>=cuz 9745   seqcseq 10699

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 2802  df-un 3202  df-in 3204  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-xp 4729  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-recs 6466  df-frec 6552  df-seqfrec 10700

This theorem is referenced by:  seq3f1olemstep  10766  seq3f1olemp  10767  nfsum1  11907  nfsum  11908  nfcprod1  12105  nfcprod  12106