Theorem nfseq 10819

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 10810	. 2 |- seq M ( .+ , F ) = ran frec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
2		nfcv 2384	. . . . . 6 |- F/_ x ZZ>=
3		nfseq.1	. . . . . 6 |- F/_ x M
4	2, 3	nffv 5680	. . . . 5 |- F/_ x ( ZZ>= ` M )
5		nfcv 2384	. . . . 5 |- F/_ x _V
6		nfcv 2384	. . . . . 6 |- F/_ x ( z + 1 )
7		nfcv 2384	. . . . . . 7 |- F/_ x w
8		nfseq.2	. . . . . . 7 |- F/_ x .+
9		nfseq.3	. . . . . . . 8 |- F/_ x F
10	9, 6	nffv 5680	. . . . . . 7 |- F/_ x ( F ` ( z + 1 ) )
11	7, 8, 10	nfov 6080	. . . . . 6 |- F/_ x ( w .+ ( F ` ( z + 1 ) ) )
12	6, 11	nfop 3899	. . . . 5 |- F/_ x <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >.
13	4, 5, 12	nfmpo 6122	. . . 4 |- F/_ x ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. )
14	9, 3	nffv 5680	. . . . 5 |- F/_ x ( F ` M )
15	3, 14	nfop 3899	. . . 4 |- F/_ x <. M , ( F ` M ) >.
16	13, 15	nffrec 6627	. . 3 |- F/_ xfrec ( ( z e. ( ZZ>= ` M ) , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
17	16	nfrn 5002	. 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 2381	1 |- F/_ x seq M ( .+ , F )

Syntax hints:   F/_wnfc 2371   _Vcvv 2813   <.cop 3692   ran crn 4750   ` cfv 5352  (class class class)co 6050   e. cmpo 6052  freccfrec 6621   1c1 8128   + caddc 8130   ZZ>=cuz 9853   seqcseq 10809

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-recs 6536  df-frec 6622  df-seqfrec 10810

This theorem is referenced by:  seq3f1olemstep  10876  seq3f1olemp  10877  nfsum1  12041  nfsum  12042  nfcprod1  12240  nfcprod  12241