Theorem iseqp1 9445

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.ex	|- ( ph -> S e. V )
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, F   x, .+ , y   x, M, y   x, N, y   x, S, y   ph, x, y

Allowed substitution hints:   V( 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 8624	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . . 4 |- ( ph -> M e. ZZ )
4		eqid 2081	. . . 4 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , M ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , M )
5		iseqp1.ex	. . . 4 |- ( ph -> S e. V )
6		uzid 8633	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
7	3, 6	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
8		iseqp1.f	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
9	8	ralrimiva 2434	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
10		fveq2 5198	. . . . . . 7 |- ( x = M -> ( F ` x ) = ( F ` M ) )
11	10	eleq1d 2147	. . . . . 6 |- ( x = M -> ( ( F ` x ) e. S <-> ( F ` M ) e. S ) )
12	11	rspcv 2697	. . . . 5 |- ( M e. ( ZZ>= ` M ) -> ( A. x e. ( ZZ>= ` M ) ( F ` x ) e. S -> ( F ` M ) e. S ) )
13	7, 9, 12	sylc 61	. . . 4 |- ( ph -> ( F ` M ) e. S )
14		iseqp1.pl	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
15	8, 14	iseqovex 9439	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. S )
16		eqid 2081	. . . 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 ) >. )
17	16, 8, 14	iseqval 9440	. . . 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 ) >. ) )
18	3, 4, 5, 13, 15, 16, 17	frecuzrdgsuc 9417	. . 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 ) ) )
19	1, 18	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 ) ) )
20	1, 5, 8, 14	iseqcl 9443	. . 3 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) e. S )
21		peano2uz 8671	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
22	1, 21	syl 14	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
23		fveq2 5198	. . . . . . 7 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
24	23	eleq1d 2147	. . . . . 6 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( N + 1 ) ) e. S ) )
25	24	rspcv 2697	. . . . 5 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( A. x e. ( ZZ>= ` M ) ( F ` x ) e. S -> ( F ` ( N + 1 ) ) e. S ) )
26	22, 9, 25	sylc 61	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. S )
27	14, 20, 26	caovcld 5674	. . 3 |- ( ph -> ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S )
28		oveq1 5539	. . . . . 6 |- ( z = N -> ( z + 1 ) = ( N + 1 ) )
29	28	fveq2d 5202	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
30	29	oveq2d 5548	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
31		oveq1 5539	. . . 4 |- ( w = ( seq M ( .+ , F , S ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
32		eqid 2081	. . . 4 |- ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) = ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) )
33	30, 31, 32	ovmpt2g 5655	. . 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 ) ) ) )
34	1, 20, 27, 33	syl3anc 1169	. 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 ) ) ) )
35	19, 34	eqtrd 2113	1 |- ( ph -> ( seq M ( .+ , F , S ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   <.cop 3401   |-> cmpt 3839   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534  freccfrec 6000   1c1 6982   + caddc 6984   ZZcz 8351   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-in1 576  ax-in2 577  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-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  iseqss  9446  iseqm1  9447  iseqfveq2  9448  iseqshft2  9452  isermono  9457  iseqsplit  9458  iseqcaopr3  9460  iseqid3s  9466  iseqhomo  9468  iseqz  9469  expivallem  9477  expp1  9483  facp1  9657  resqrexlemfp1  9895  climserile  10183  ialgrp1  10428