Theorem nfseq 10549

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 10540	. 2 |- seq M ( .+ , F ) = ran frec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
2		nfcv 2339	. . . . . 6 |- F/_ x ZZ>=
3		nfseq.1	. . . . . 6 |- F/_ x M
4	2, 3	nffv 5568	. . . . 5 |- F/_ x ( ZZ>= ` M )
5		nfcv 2339	. . . . 5 |- F/_ x _V
6		nfcv 2339	. . . . . 6 |- F/_ x ( z + 1 )
7		nfcv 2339	. . . . . . 7 |- F/_ x w
8		nfseq.2	. . . . . . 7 |- F/_ x .+
9		nfseq.3	. . . . . . . 8 |- F/_ x F
10	9, 6	nffv 5568	. . . . . . 7 |- F/_ x ( F ` ( z + 1 ) )
11	7, 8, 10	nfov 5952	. . . . . 6 |- F/_ x ( w .+ ( F ` ( z + 1 ) ) )
12	6, 11	nfop 3824	. . . . 5 |- F/_ x <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >.
13	4, 5, 12	nfmpo 5991	. . . 4 |- F/_ x ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. )
14	9, 3	nffv 5568	. . . . 5 |- F/_ x ( F ` M )
15	3, 14	nfop 3824	. . . 4 |- F/_ x <. M , ( F ` M ) >.
16	13, 15	nffrec 6454	. . 3 |- F/_ xfrec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
17	16	nfrn 4911	. 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 2336	1 |- F/_ x seq M ( .+ , F )

Syntax hints:   F/_wnfc 2326   _Vcvv 2763   <.cop 3625   ran crn 4664   ` cfv 5258  (class class class)co 5922   e. cmpo 5924  freccfrec 6448   1c1 7880   + caddc 7882   ZZ>=cuz 9601   seqcseq 10539

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-recs 6363  df-frec 6449  df-seqfrec 10540

This theorem is referenced by:  seq3f1olemstep  10606  seq3f1olemp  10607  nfsum1  11521  nfsum  11522  nfcprod1  11719  nfcprod  11720