Theorem nfseq 10600

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 10591	. 2 |- seq M ( .+ , F ) = ran frec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
2		nfcv 2347	. . . . . 6 |- F/_ x ZZ>=
3		nfseq.1	. . . . . 6 |- F/_ x M
4	2, 3	nffv 5585	. . . . 5 |- F/_ x ( ZZ>= ` M )
5		nfcv 2347	. . . . 5 |- F/_ x _V
6		nfcv 2347	. . . . . 6 |- F/_ x ( z + 1 )
7		nfcv 2347	. . . . . . 7 |- F/_ x w
8		nfseq.2	. . . . . . 7 |- F/_ x .+
9		nfseq.3	. . . . . . . 8 |- F/_ x F
10	9, 6	nffv 5585	. . . . . . 7 |- F/_ x ( F ` ( z + 1 ) )
11	7, 8, 10	nfov 5973	. . . . . 6 |- F/_ x ( w .+ ( F ` ( z + 1 ) ) )
12	6, 11	nfop 3834	. . . . 5 |- F/_ x <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >.
13	4, 5, 12	nfmpo 6013	. . . 4 |- F/_ x ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. )
14	9, 3	nffv 5585	. . . . 5 |- F/_ x ( F ` M )
15	3, 14	nfop 3834	. . . 4 |- F/_ x <. M , ( F ` M ) >.
16	13, 15	nffrec 6481	. . 3 |- F/_ xfrec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
17	16	nfrn 4922	. 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 2344	1 |- F/_ x seq M ( .+ , F )

Syntax hints:   F/_wnfc 2334   _Vcvv 2771   <.cop 3635   ran crn 4675   ` cfv 5270  (class class class)co 5943   e. cmpo 5945  freccfrec 6475   1c1 7925   + caddc 7927   ZZ>=cuz 9647   seqcseq 10590

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-un 3169  df-in 3171  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-xp 4680  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-recs 6390  df-frec 6476  df-seqfrec 10591

This theorem is referenced by:  seq3f1olemstep  10657  seq3f1olemp  10658  nfsum1  11609  nfsum  11610  nfcprod1  11807  nfcprod  11808