Theorem iseqp1 9882

Description: Value of the sequence builder function at a successor.

New proofs should use seq3p1 9884 instead (together with iseqsst 9886 or iseqseq3 9902 if need be).

(Contributed by Jim Kingdon, 31-May-2020.) (New usage is discouraged.)

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 9024	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . . 4 |- ( ph -> M e. ZZ )
4		eqid 2088	. . . 4 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , M ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , M )
5		fveq2 5305	. . . . . 6 |- ( x = M -> ( F ` x ) = ( F ` M ) )
6	5	eleq1d 2156	. . . . 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 2446	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
9		uzid 9033	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
10	3, 9	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
11	6, 8, 10	rspcdva 2727	. . . 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 9870	. . . 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 2088	. . . 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 9871	. . . 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 9821	. . 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 9881	. . 3 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) e. S )
19		fveq2 5305	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
20	19	eleq1d 2156	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( N + 1 ) ) e. S ) )
21		peano2uz 9071	. . . . . 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 2727	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. S )
24	12, 18, 23	caovcld 5798	. . 3 |- ( ph -> ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S )
25		oveq1 5659	. . . . . 6 |- ( z = N -> ( z + 1 ) = ( N + 1 ) )
26	25	fveq2d 5309	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
27	26	oveq2d 5668	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
28		oveq1 5659	. . . 4 |- ( w = ( seq M ( .+ , F , S ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
29		eqid 2088	. . . 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 5779	. . 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 1174	. 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 2120	1 |- ( ph -> ( seq M ( .+ , F , S ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   <.cop 3449   |-> cmpt 3899   ` cfv 5015  (class class class)co 5652   |-> cmpt2 5654  freccfrec 6155   1c1 7351   + caddc 7353   ZZcz 8750   ZZ>=cuz 9019   seqcseq4 9851

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-iseq 9853

This theorem is referenced by:  iseqoveq  9885  iseqsst  9886  iseqm1  9888  iseqfveq2  9890  iseqshft2  9898  isermono  9906  iseqsplit  9908  iseqcaopr3  9910  iseqid3s  9938  iseqid2  9941  iseqhomo  9942  iseqz  9943  facp1  10138  iseqcoll  10247  ialgrp1  11306