Theorem seq3p1 10461

Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)

Hypotheses

Ref	Expression
seq3p1.m	|- ( ph -> N e. ( ZZ>= ` M ) )
seq3p1.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
seq3p1.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
seq3p1	|- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Distinct variable groups:   x, .+ , y   x, F, y   x, M, y   x, N, y   x, S, y   ph, x, y

Proof of Theorem seq3p1

Dummy variables a b w z c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seq3p1.m	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzel2 9532	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . . 4 |- ( ph -> M e. ZZ )
4		fveq2 5515	. . . . . 6 |- ( x = M -> ( F ` x ) = ( F ` M ) )
5	4	eleq1d 2246	. . . . 5 |- ( x = M -> ( ( F ` x ) e. S <-> ( F ` M ) e. S ) )
6		seq3p1.f	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
7	6	ralrimiva 2550	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
8		uzid 9541	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
9	3, 8	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
10	5, 7, 9	rspcdva 2846	. . . 4 |- ( ph -> ( F ` M ) e. S )
11		ssv 3177	. . . . 5 |- S C_ _V
12	11	a1i 9	. . . 4 |- ( ph -> S C_ _V )
13		seq3p1.pl	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
14	6, 13	iseqovex 10455	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. S )
15		iseqvalcbv 10456	. . . 4 |- frec ( ( a e. ( ZZ>= ` M ) , b e. _V |-> <. ( a + 1 ) , ( a ( c e. ( ZZ>= ` M ) , d e. S |-> ( d .+ ( F ` ( c + 1 ) ) ) ) b ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )
16	3, 15, 6, 13	seq3val 10457	. . . 4 |- ( ph -> seq M ( .+ , F ) = ran frec ( ( a e. ( ZZ>= ` M ) , b e. _V |-> <. ( a + 1 ) , ( a ( c e. ( ZZ>= ` M ) , d e. S |-> ( d .+ ( F ` ( c + 1 ) ) ) ) b ) >. ) , <. M , ( F ` M ) >. ) )
17	3, 10, 12, 14, 15, 16	frecuzrdgsuct 10423	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F ) ` N ) ) )
18	1, 17	mpdan 421	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F ) ` N ) ) )
19		eqid 2177	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
20	19, 3, 6, 13	seqf 10460	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
21	20, 1	ffvelcdmd 5652	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. S )
22		fveq2 5515	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
23	22	eleq1d 2246	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( N + 1 ) ) e. S ) )
24		peano2uz 9582	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
25	1, 24	syl 14	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
26	23, 7, 25	rspcdva 2846	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. S )
27	13, 21, 26	caovcld 6027	. . 3 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S )
28		fvoveq1 5897	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
29	28	oveq2d 5890	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
30		oveq1 5881	. . . 4 |- ( w = ( seq M ( .+ , F ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
31		eqid 2177	. . . 4 |- ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) = ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) )
32	29, 30, 31	ovmpog 6008	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( seq M ( .+ , F ) ` N ) e. S /\ ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S ) -> ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F ) ` N ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
33	1, 21, 27, 32	syl3anc 1238	. 2 |- ( ph -> ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F ) ` N ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
34	18, 33	eqtrd 2210	1 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2737   C_ wss 3129   <.cop 3595   ` cfv 5216  (class class class)co 5874   e. cmpo 5876  freccfrec 6390   1c1 7811   + caddc 7813   ZZcz 9252   ZZ>=cuz 9527   seqcseq 10444

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253  df-uz 9528  df-seqfrec 10445

This theorem is referenced by:  seq3clss  10466  seq3m1  10467  seq3fveq2  10468  seq3shft2  10472  ser3mono  10477  seq3split  10478  seq3caopr3  10480  seq3id3  10506  seq3id2  10508  seq3homo  10509  seq3z  10510  ser3ge0  10516  exp3vallem  10520  expp1  10526  facp1  10709  seq3coll  10821  resqrexlemfp1  11017  climserle  11352  clim2prod  11546  prodfap0  11552  prodfrecap  11553  ege2le3  11678  efgt1p2  11702  efgt1p  11703  algrp1  12045  pcmpt  12340  nninfdclemp1  12450  mulgnnp1  12990