Theorem seq3p1 10418

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 9492	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . . 4 |- ( ph -> M e. ZZ )
4		fveq2 5496	. . . . . 6 |- ( x = M -> ( F ` x ) = ( F ` M ) )
5	4	eleq1d 2239	. . . . 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 2543	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
8		uzid 9501	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
9	3, 8	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
10	5, 7, 9	rspcdva 2839	. . . 4 |- ( ph -> ( F ` M ) e. S )
11		ssv 3169	. . . . 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 10412	. . . 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 10413	. . . 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 10414	. . . 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 10380	. . 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 419	. 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 2170	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
20	19, 3, 6, 13	seqf 10417	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
21	20, 1	ffvelrnd 5632	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. S )
22		fveq2 5496	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
23	22	eleq1d 2239	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( N + 1 ) ) e. S ) )
24		peano2uz 9542	. . . . . 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 2839	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. S )
27	13, 21, 26	caovcld 6006	. . 3 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S )
28		fvoveq1 5876	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
29	28	oveq2d 5869	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
30		oveq1 5860	. . . 4 |- ( w = ( seq M ( .+ , F ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
31		eqid 2170	. . . 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 5987	. . 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 1233	. 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 2203	1 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   <.cop 3586   ` cfv 5198  (class class class)co 5853   e. cmpo 5855  freccfrec 6369   1c1 7775   + caddc 7777   ZZcz 9212   ZZ>=cuz 9487   seqcseq 10401

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402

This theorem is referenced by:  seq3clss  10423  seq3m1  10424  seq3fveq2  10425  seq3shft2  10429  ser3mono  10434  seq3split  10435  seq3caopr3  10437  seq3id3  10463  seq3id2  10465  seq3homo  10466  seq3z  10467  ser3ge0  10473  exp3vallem  10477  expp1  10483  facp1  10664  seq3coll  10777  resqrexlemfp1  10973  climserle  11308  clim2prod  11502  prodfap0  11508  prodfrecap  11509  ege2le3  11634  efgt1p2  11658  efgt1p  11659  algrp1  12000  pcmpt  12295  nninfdclemp1  12405