Theorem seqp1cd 10795

Description: Value of the sequence builder function at a successor. A version of seq3p1 10790 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 9821	. . . . 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 3250	. . . . 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 10792	. . . 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 10784	. . . 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 10786	. . . 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 10749	. . 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 2231	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
15	4, 8, 14, 3, 7	seqf2 10793	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> C )
16	15, 1	ffvelcdmd 5791	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. C )
17		fveq2 5648	. . . . . 6 |- ( x = ( N + 1 ) -> ( F ` x ) = ( F ` ( N + 1 ) ) )
18	17	eleq1d 2300	. . . . 5 |- ( x = ( N + 1 ) -> ( ( F ` x ) e. D <-> ( F ` ( N + 1 ) ) e. D ) )
19	7	ralrimiva 2606	. . . . 5 |- ( ph -> A. x e. ( ZZ>= ` ( M + 1 ) ) ( F ` x ) e. D )
20		eluzp1p1 9843	. . . . . 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 2916	. . . 4 |- ( ph -> ( F ` ( N + 1 ) ) e. D )
23	8, 16, 22	caovcld 6186	. . 3 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) e. C )
24		fvoveq1 6051	. . . . 5 |- ( z = N -> ( F ` ( z + 1 ) ) = ( F ` ( N + 1 ) ) )
25	24	oveq2d 6044	. . . 4 |- ( z = N -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( N + 1 ) ) ) )
26		oveq1 6035	. . . 4 |- ( w = ( seq M ( .+ , F ) ` N ) -> ( w .+ ( F ` ( N + 1 ) ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
27		eqid 2231	. . . 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 6166	. . 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 1274	. 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 2264	1 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   C_ wss 3201   <.cop 3676   ` cfv 5333  (class class class)co 6028   e. cmpo 6030  freccfrec 6599   1c1 8093   + caddc 8095   ZZcz 9540   ZZ>=cuz 9816   seqcseq 10772

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-seqfrec 10773

This theorem is referenced by:  ennnfonelemp1  13107