Theorem iseqp1t 9609

Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)

Hypotheses

Ref	Expression
iseqp1t.m	|- ( ph -> N e. ( ZZ>= ` M ) )
iseqp1t.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqp1t.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
iseqp1t.t	|- ( ph -> S C_ T )

Assertion

Ref	Expression
iseqp1t	|- ( ph -> ( seq M ( .+ , F , T ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F , T ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Distinct variable groups:   x, .+ , y   x, F, y   x, M, y   x, N, y   x, S, y   x, T, y   ph, x, y

Proof of Theorem iseqp1t

Dummy variables a b w z c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqp1t.m	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzel2 8775	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . . 4 |- ( ph -> M e. ZZ )
4		fveq2 5230	. . . . . 6 |- ( x = M -> ( F ` x ) = ( F ` M ) )
5	4	eleq1d 2151	. . . . 5 |- ( x = M -> ( ( F ` x ) e. S <-> ( F ` M ) e. S ) )
6		iseqp1t.f	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
7	6	ralrimiva 2439	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
8		uzid 8784	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
9	3, 8	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
10	5, 7, 9	rspcdva 2715	. . . 4 |- ( ph -> ( F ` M ) e. S )
11		iseqp1t.t	. . . 4 |- ( ph -> S C_ T )
12		iseqp1t.pl	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
13	6, 12	iseqovex 9599	. . . 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		iseqvalcbv 9601	. . . 4 |- frec ( ( a e. ( ZZ>= ` M ) , b e. T |-> <. ( 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. T |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )
15	3, 14, 6, 12, 11	iseqvalt 9602	. . . 4 |- ( ph -> seq M ( .+ , F , T ) = ran frec ( ( a e. ( ZZ>= ` M ) , b e. T |-> <. ( a + 1 ) , ( a ( c e. ( ZZ>= ` M ) , d e. S |-> ( d .+ ( F ` ( c + 1 ) ) ) ) b ) >. ) , <. M , ( F ` M ) >. ) )
16	3, 10, 11, 13, 14, 15	frecuzrdgsuct 9576	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , T ) ` ( N + 1 ) ) = ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F , T ) ` N ) ) )
17	1, 16	mpdan 412	. 2 |- ( ph -> ( seq M ( .+ , F , T ) ` ( N + 1 ) ) = ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F , T ) ` N ) ) )
18		eqid 2083	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
19	18, 3, 6, 12, 11	iseqfclt 9606	. . . 4 |- ( ph -> seq M ( .+ , F , T ) : ( ZZ>= ` M ) --> S )
20	19, 1	ffvelrnd 5356	. . 3 |- ( ph -> ( seq M ( .+ , F , T ) ` N ) e. S )
21		fveq2 5230	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
22	21	eleq1d 2151	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( N + 1 ) ) e. S ) )
23		peano2uz 8822	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
24	1, 23	syl 14	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
25	22, 7, 24	rspcdva 2715	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. S )
26	12, 20, 25	caovcld 5706	. . 3 |- ( ph -> ( ( seq M ( .+ , F , T ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S )
27		oveq1 5571	. . . . . 6 |- ( z = N -> ( z + 1 ) = ( N + 1 ) )
28	27	fveq2d 5234	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
29	28	oveq2d 5580	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
30		oveq1 5571	. . . 4 |- ( w = ( seq M ( .+ , F , T ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F , T ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
31		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 ) ) ) )
32	29, 30, 31	ovmpt2g 5687	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( seq M ( .+ , F , T ) ` N ) e. S /\ ( ( seq M ( .+ , F , T ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. S ) -> ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F , T ) ` N ) ) = ( ( seq M ( .+ , F , T ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
33	1, 20, 26, 32	syl3anc 1170	. 2 |- ( ph -> ( N ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F , T ) ` N ) ) = ( ( seq M ( .+ , F , T ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
34	17, 33	eqtrd 2115	1 |- ( ph -> ( seq M ( .+ , F , T ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F , T ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   C_ wss 2982   <.cop 3419   ` cfv 4952  (class class class)co 5564   |-> cmpt2 5566  freccfrec 6060   1c1 7114   + caddc 7116   ZZcz 8502   ZZ>=cuz 8770   seqcseq 9591

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 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-addcom 7208  ax-addass 7210  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-0id 7216  ax-rnegex 7217  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-ltadd 7224

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 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771  df-iseq 9592

This theorem is referenced by:  iseqsst  9612