Theorem seqp1cd 10469

Description: Value of the sequence builder function at a successor. A version of seq3p1 10465 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 9536	. . . . 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 3179	. . . . 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 10466	. . . 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 10460	. . . 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 10462	. . . 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 10427	. . 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 2177	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
15	4, 8, 14, 3, 7	seqf2 10467	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> C )
16	15, 1	ffvelcdmd 5655	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. C )
17		fveq2 5517	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
18	17	eleq1d 2246	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. D <-> ( F ` ( N + 1 ) ) e. D ) )
19	7	ralrimiva 2550	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` ( M + 1 ) ) ( F ` x ) e. D )
20		eluzp1p1 9556	. . . . . 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 2848	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. D )
23	8, 16, 22	caovcld 6031	. . 3 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. C )
24		fvoveq1 5901	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
25	24	oveq2d 5894	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
26		oveq1 5885	. . . 4 |- ( w = ( seq M ( .+ , F ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
27		eqid 2177	. . . 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 6012	. . 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 1238	. 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 2210	1 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   C_ wss 3131   <.cop 3597   ` cfv 5218  (class class class)co 5878   e. cmpo 5880  freccfrec 6394   1c1 7815   + caddc 7817   ZZcz 9256   ZZ>=cuz 9531   seqcseq 10448

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257  df-uz 9532  df-seqfrec 10449

This theorem is referenced by:  ennnfonelemp1  12410