Theorem nfseq 12811

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-seq 12802	. 2 |- seq M ( .+ , F ) = ( rec ( ( z e. _V , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om )
2		nfcv 2764	. . . . 5 |- F/_ x _V
3		nfcv 2764	. . . . . 6 |- F/_ x ( z + 1 )
4		nfcv 2764	. . . . . . 7 |- F/_ x w
5		nfseq.2	. . . . . . 7 |- F/_ x .+
6		nfseq.3	. . . . . . . 8 |- F/_ x F
7	6, 3	nffv 6198	. . . . . . 7 |- F/_ x ( F ` ( z + 1 ) )
8	4, 5, 7	nfov 6676	. . . . . 6 |- F/_ x ( w .+ ( F ` ( z + 1 ) ) )
9	3, 8	nfop 4418	. . . . 5 |- F/_ x <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >.
10	2, 2, 9	nfmpt2 6724	. . . 4 |- F/_ x ( z e. _V , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. )
11		nfseq.1	. . . . 5 |- F/_ x M
12	6, 11	nffv 6198	. . . . 5 |- F/_ x ( F ` M )
13	11, 12	nfop 4418	. . . 4 |- F/_ x <. M , ( F ` M ) >.
14	10, 13	nfrdg 7510	. . 3 |- F/_ x rec ( ( z e. _V , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
15		nfcv 2764	. . 3 |- F/_ x om
16	14, 15	nfima 5474	. 2 |- F/_ x ( rec ( ( z e. _V , w e. _V |-> <. ( z + 1 ) , ( w .+ ( F ` ( z + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om )
17	1, 16	nfcxfr 2762	1 |- F/_ x seq M ( .+ , F )

Syntax hints:   F/_wnfc 2751   _Vcvv 3200   <.cop 4183   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   reccrdg 7505   1c1 9937   + caddc 9939   seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802

This theorem is referenced by:  seqof2  12859  nfsum1  14420  nfsum  14421  nfcprod1  14640  nfcprod  14641  lgamgulm2  24762  binomcxplemdvbinom  38552  binomcxplemdvsum  38554  binomcxplemnotnn0  38555  fmuldfeqlem1  39814  fmuldfeq  39815  sumnnodd  39862  stoweidlem51  40268