Theorem seqp1cd 10505

Description: Value of the sequence builder function at a successor. A version of seq3p1 10501 which provides two classes D and C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)

Hypotheses

Ref	Expression
seqp1cd.m	|- ( ph -> N e. ( ZZ>= ` M ) )
seqp1cd.1	|- ( ph -> ( F ` M ) e. C )
seqp1cd.2	|- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )
seqp1cd.5	|- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )

Assertion

Ref	Expression
seqp1cd	|- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Distinct variable groups:   x, .+ , y   x, C, y   x, D, y   x, F, y   x, M, y   x, N, y   ph, x, y

Proof of Theorem seqp1cd

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

Step	Hyp	Ref	Expression
1		seqp1cd.m	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzel2 9568	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . . 4 |- ( ph -> M e. ZZ )
4		seqp1cd.1	. . . 4 |- ( ph -> ( F ` M ) e. C )
5		ssv 3192	. . . . 5 |- C C_ _V
6	5	a1i 9	. . . 4 |- ( ph -> C C_ _V )
7		seqp1cd.5	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )
8		seqp1cd.2	. . . . 5 |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )
9	7, 8	seqovcd 10502	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. C ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. C )
10		iseqvalcbv 10496	. . . 4 |- frec ( ( a e. ( ZZ>= ` M ) , b e. _V |-> <. ( a + 1 ) , ( a ( c e. ( ZZ>= ` M ) , d e. C |-> ( 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. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )
11	3, 10, 4, 8, 7	seqvalcd 10498	. . . 4 |- ( ph -> seq M ( .+ , F ) = ran frec ( ( a e. ( ZZ>= ` M ) , b e. _V |-> <. ( a + 1 ) , ( a ( c e. ( ZZ>= ` M ) , d e. C |-> ( d .+ ( F ` ( c + 1 ) ) ) ) b ) >. ) , <. M , ( F ` M ) >. ) )
12	3, 4, 6, 9, 10, 11	frecuzrdgsuct 10461	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( N ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F ) ` N ) ) )
13	1, 12	mpdan 421	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( N ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F ) ` N ) ) )
14		eqid 2189	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
15	4, 8, 14, 3, 7	seqf2 10503	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> C )
16	15, 1	ffvelcdmd 5676	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. C )
17		fveq2 5537	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
18	17	eleq1d 2258	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. D <-> ( F ` ( N + 1 ) ) e. D ) )
19	7	ralrimiva 2563	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` ( M + 1 ) ) ( F ` x ) e. D )
20		eluzp1p1 9589	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` ( M + 1 ) ) )
21	1, 20	syl 14	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` ( M + 1 ) ) )
22	18, 19, 21	rspcdva 2861	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. D )
23	8, 16, 22	caovcld 6054	. . 3 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. C )
24		fvoveq1 5923	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
25	24	oveq2d 5916	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
26		oveq1 5907	. . . 4 |- ( w = ( seq M ( .+ , F ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
27		eqid 2189	. . . 4 |- ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) = ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) )
28	25, 26, 27	ovmpog 6035	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( seq M ( .+ , F ) ` N ) e. C /\ ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. C ) -> ( N ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F ) ` N ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
29	1, 16, 23, 28	syl3anc 1249	. 2 |- ( ph -> ( N ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) ( seq M ( .+ , F ) ` N ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
30	13, 29	eqtrd 2222	1 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   C_ wss 3144   <.cop 3613   ` cfv 5238  (class class class)co 5900   e. cmpo 5902  freccfrec 6419   1c1 7847   + caddc 7849   ZZcz 9288   ZZ>=cuz 9563   seqcseq 10484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-ltadd 7962

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-inn 8955  df-n0 9212  df-z 9289  df-uz 9564  df-seqfrec 10485

This theorem is referenced by:  ennnfonelemp1  12468