Theorem seq3p1 10393

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 9467	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . . 4 |- ( ph -> M e. ZZ )
4		fveq2 5485	. . . . . 6 |- ( x = M -> ( F ` x ) = ( F ` M ) )
5	4	eleq1d 2234	. . . . 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 2538	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
8		uzid 9476	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
9	3, 8	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
10	5, 7, 9	rspcdva 2834	. . . 4 |- ( ph -> ( F ` M ) e. S )
11		ssv 3163	. . . . 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 10387	. . . 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 10388	. . . 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 10389	. . . 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 10355	. . 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 418	. 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 2165	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
20	19, 3, 6, 13	seqf 10392	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
21	20, 1	ffvelrnd 5620	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. S )
22		fveq2 5485	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
23	22	eleq1d 2234	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( N + 1 ) ) e. S ) )
24		peano2uz 9517	. . . . . 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 2834	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. S )
27	13, 21, 26	caovcld 5991	. . 3 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S )
28		fvoveq1 5864	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
29	28	oveq2d 5857	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
30		oveq1 5848	. . . 4 |- ( w = ( seq M ( .+ , F ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
31		eqid 2165	. . . 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 5972	. . 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 1228	. 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 2198	1 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2725   C_ wss 3115   <.cop 3578   ` cfv 5187  (class class class)co 5841   e. cmpo 5843  freccfrec 6354   1c1 7750   + caddc 7752   ZZcz 9187   ZZ>=cuz 9462   seqcseq 10376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-addcom 7849  ax-addass 7851  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-0id 7857  ax-rnegex 7858  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-ltadd 7865

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-inn 8854  df-n0 9111  df-z 9188  df-uz 9463  df-seqfrec 10377

This theorem is referenced by:  seq3clss  10398  seq3m1  10399  seq3fveq2  10400  seq3shft2  10404  ser3mono  10409  seq3split  10410  seq3caopr3  10412  seq3id3  10438  seq3id2  10440  seq3homo  10441  seq3z  10442  ser3ge0  10448  exp3vallem  10452  expp1  10458  facp1  10639  seq3coll  10751  resqrexlemfp1  10947  climserle  11282  clim2prod  11476  prodfap0  11482  prodfrecap  11483  ege2le3  11608  efgt1p2  11632  efgt1p  11633  algrp1  11974  pcmpt  12269  nninfdclemp1  12379