Theorem nfseq 10765

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 10756	. 2 |- seq M ( .+ , F ) = ran frec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
2		nfcv 2375	. . . . . 6 |- F/_ x ZZ>=
3		nfseq.1	. . . . . 6 |- F/_ x M
4	2, 3	nffv 5658	. . . . 5 |- F/_ x ( ZZ>= ` M )
5		nfcv 2375	. . . . 5 |- F/_ x _V
6		nfcv 2375	. . . . . 6 |- F/_ x ( z + 1 )
7		nfcv 2375	. . . . . . 7 |- F/_ x w
8		nfseq.2	. . . . . . 7 |- F/_ x .+
9		nfseq.3	. . . . . . . 8 |- F/_ x F
10	9, 6	nffv 5658	. . . . . . 7 |- F/_ x ( F ` ( z + 1 ) )
11	7, 8, 10	nfov 6058	. . . . . 6 |- F/_ x ( w .+ ( F ` ( z + 1 ) ) )
12	6, 11	nfop 3883	. . . . 5 |- F/_ x <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >.
13	4, 5, 12	nfmpo 6100	. . . 4 |- F/_ x ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. )
14	9, 3	nffv 5658	. . . . 5 |- F/_ x ( F ` M )
15	3, 14	nfop 3883	. . . 4 |- F/_ x <. M , ( F ` M ) >.
16	13, 15	nffrec 6605	. . 3 |- F/_ xfrec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
17	16	nfrn 4983	. 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 2372	1 |- F/_ x seq M ( .+ , F )

Syntax hints:   F/_wnfc 2362   _Vcvv 2803   <.cop 3676   ran crn 4732   ` cfv 5333  (class class class)co 6028   e. cmpo 6030  freccfrec 6599   1c1 8076   + caddc 8078   ZZ>=cuz 9799   seqcseq 10755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-in 3207  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-recs 6514  df-frec 6600  df-seqfrec 10756

This theorem is referenced by:  seq3f1olemstep  10822  seq3f1olemp  10823  nfsum1  11979  nfsum  11980  nfcprod1  12178  nfcprod  12179