Theorem nfiseq 9438

Description: Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)

Hypotheses

Ref	Expression
nfiseq.1	|- F/_ x M
nfiseq.2	|- F/_ x .+
nfiseq.3	|- F/_ x F
nfiseq.4	|- F/_ x S

Assertion

Ref	Expression
nfiseq	|- F/_ x seq M ( .+ , F , S )

Proof of Theorem nfiseq

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iseq 9432	. 2 |- seq M ( .+ , F , S ) = ran frec ( ( y e. ( ZZ>= ` M ) , z e. S |-> <. ( y + 1 ) , ( z .+ ( F ` ( y + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
2		nfcv 2219	. . . . . 6 |- F/_ x ZZ>=
3		nfiseq.1	. . . . . 6 |- F/_ x M
4	2, 3	nffv 5205	. . . . 5 |- F/_ x ( ZZ>= ` M )
5		nfiseq.4	. . . . 5 |- F/_ x S
6		nfcv 2219	. . . . . 6 |- F/_ x ( y + 1 )
7		nfcv 2219	. . . . . . 7 |- F/_ x z
8		nfiseq.2	. . . . . . 7 |- F/_ x .+
9		nfiseq.3	. . . . . . . 8 |- F/_ x F
10	9, 6	nffv 5205	. . . . . . 7 |- F/_ x ( F ` ( y + 1 ) )
11	7, 8, 10	nfov 5555	. . . . . 6 |- F/_ x ( z .+ ( F ` ( y + 1 ) ) )
12	6, 11	nfop 3586	. . . . 5 |- F/_ x <. ( y + 1 ) , ( z .+ ( F ` ( y + 1 ) ) ) >.
13	4, 5, 12	nfmpt2 5593	. . . 4 |- F/_ x ( y e. ( ZZ>= ` M ) , z e. S |-> <. ( y + 1 ) , ( z .+ ( F ` ( y + 1 ) ) ) >. )
14	9, 3	nffv 5205	. . . . 5 |- F/_ x ( F ` M )
15	3, 14	nfop 3586	. . . 4 |- F/_ x <. M , ( F ` M ) >.
16	13, 15	nffrec 6005	. . 3 |- F/_ xfrec ( ( y e. ( ZZ>= ` M ) , z e. S |-> <. ( y + 1 ) , ( z .+ ( F ` ( y + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
17	16	nfrn 4597	. 2 |- F/_ x ran frec ( ( y e. ( ZZ>= ` M ) , z e. S |-> <. ( y + 1 ) , ( z .+ ( F ` ( y + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
18	1, 17	nfcxfr 2216	1 |- F/_ x seq M ( .+ , F , S )

Syntax hints:   F/_wnfc 2206   <.cop 3401   ran crn 4364   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534  freccfrec 6000   1c1 6982   + caddc 6984   ZZ>=cuz 8619   seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-recs 5943  df-frec 6001  df-iseq 9432

This theorem is referenced by:  nfsum1  10193  nfsum  10194