Theorem iseqp1 9757

Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)

Hypotheses

Ref	Expression
iseqp1.m	|- ( ph -> N e. ( ZZ>= ` M ) )
iseqp1.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqp1.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
iseqp1	|- ( ph -> ( seq M ( .+ , F , S ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F , S ) ` 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 iseqp1

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

Step	Hyp	Ref	Expression
1		iseqp1.m	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzel2 8919	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . . 4 |- ( ph -> M e. ZZ )
4		eqid 2083	. . . 4 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , M ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , M )
5		fveq2 5253	. . . . . 6 |- ( x = M -> ( F ` x ) = ( F ` M ) )
6	5	eleq1d 2151	. . . . 5 |- ( x = M -> ( ( F ` x ) e. S <-> ( F ` M ) e. S ) )
7		iseqp1.f	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
8	7	ralrimiva 2440	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
9		uzid 8928	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
10	3, 9	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
11	6, 8, 10	rspcdva 2717	. . . 4 |- ( ph -> ( F ` M ) e. S )
12		iseqp1.pl	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
13	7, 12	iseqovex 9748	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. S )
14		eqid 2083	. . . 4 |- frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )
15	14, 7, 12	iseqval 9749	. . . 4 |- ( ph -> seq M ( .+ , F , S ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) )
16	3, 4, 11, 13, 14, 15	frecuzrdgsuc 9710	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` ( N + 1 ) ) = ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F , S ) ` N ) ) )
17	1, 16	mpdan 412	. 2 |- ( ph -> ( seq M ( .+ , F , S ) ` ( N + 1 ) ) = ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F , S ) ` N ) ) )
18	1, 7, 12	iseqcl 9756	. . 3 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) e. S )
19		fveq2 5253	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
20	19	eleq1d 2151	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( N + 1 ) ) e. S ) )
21		peano2uz 8966	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
22	1, 21	syl 14	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
23	20, 8, 22	rspcdva 2717	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. S )
24	12, 18, 23	caovcld 5733	. . 3 |- ( ph -> ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S )
25		oveq1 5598	. . . . . 6 |- ( z = N -> ( z + 1 ) = ( N + 1 ) )
26	25	fveq2d 5257	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
27	26	oveq2d 5607	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
28		oveq1 5598	. . . 4 |- ( w = ( seq M ( .+ , F , S ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
29		eqid 2083	. . . 4 |- ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) = ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) )
30	27, 28, 29	ovmpt2g 5714	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( seq M ( .+ , F , S ) ` N ) e. S /\ ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S ) -> ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F , S ) ` N ) ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
31	1, 18, 24, 30	syl3anc 1170	. 2 |- ( ph -> ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F , S ) ` N ) ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
32	17, 31	eqtrd 2115	1 |- ( ph -> ( seq M ( .+ , F , S ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   <.cop 3425   |-> cmpt 3865   ` cfv 4969  (class class class)co 5591   |-> cmpt2 5593  freccfrec 6087   1c1 7254   + caddc 7256   ZZcz 8646   ZZ>=cuz 8914   seqcseq 9740

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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-ltadd 7364

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-inn 8317  df-n0 8566  df-z 8647  df-uz 8915  df-iseq 9741

This theorem is referenced by:  iseqoveq  9759  iseqss  9760  iseqsst  9761  iseqm1  9762  iseqfveq2  9763  iseqshft2  9767  isermono  9772  iseqsplit  9773  iseqcaopr3  9775  iseqid3s  9781  iseqid2  9783  iseqhomo  9784  iseqz  9785  expivallem  9793  expp1  9799  facp1  9973  resqrexlemfp1  10269  climserile  10557  ialgrp1  10808