Theorem seqf 10716

Description: Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypotheses

Ref	Expression
seqf.1	|- Z = ( ZZ>= ` M )
seqf.2	|- ( ph -> M e. ZZ )
seqf.3	|- ( ( ph /\ x e. Z ) -> ( F ` x ) e. S )
seqf.4	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
seqf	|- ( ph -> seq M ( .+ , F ) : Z --> S )

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

Allowed substitution hint:   Z( y)

Proof of Theorem seqf

Dummy variables a b s t w z u v c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqf.2	. . 3 |- ( ph -> M e. ZZ )
2		fveq2 5635	. . . . 5 |- ( x = M -> ( F ` x ) = ( F ` M ) )
3	2	eleq1d 2298	. . . 4 |- ( x = M -> ( ( F ` x ) e. S <-> ( F ` M ) e. S ) )
4		seqf.3	. . . . 5 |- ( ( ph /\ x e. Z ) -> ( F ` x ) e. S )
5	4	ralrimiva 2603	. . . 4 |- ( ph -> A. x e. Z ( F ` x ) e. S )
6		uzid 9760	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
7	1, 6	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
8		seqf.1	. . . . 5 |- Z = ( ZZ>= ` M )
9	7, 8	eleqtrrdi 2323	. . . 4 |- ( ph -> M e. Z )
10	3, 5, 9	rspcdva 2913	. . 3 |- ( ph -> ( F ` M ) e. S )
11		ssv 3247	. . . 4 |- S C_ _V
12	11	a1i 9	. . 3 |- ( ph -> S C_ _V )
13		simprl 529	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> x e. ( ZZ>= ` M ) )
14		simprr 531	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> y e. S )
15		seqf.4	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
16	15	caovclg 6170	. . . . . . 7 |- ( ( ph /\ ( a e. S /\ b e. S ) ) -> ( a .+ b ) e. S )
17	16	adantlr 477	. . . . . 6 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( a e. S /\ b e. S ) ) -> ( a .+ b ) e. S )
18		fveq2 5635	. . . . . . . 8 |- ( c = ( x + 1 ) -> ( F ` c ) = ( F ` ( x + 1 ) ) )
19	18	eleq1d 2298	. . . . . . 7 |- ( c = ( x + 1 ) -> ( ( F ` c ) e. S <-> ( F ` ( x + 1 ) ) e. S ) )
20		fveq2 5635	. . . . . . . . . . 11 |- ( x = c -> ( F ` x ) = ( F ` c ) )
21	20	eleq1d 2298	. . . . . . . . . 10 |- ( x = c -> ( ( F ` x ) e. S <-> ( F ` c ) e. S ) )
22	21	cbvralv 2765	. . . . . . . . 9 |- ( A. x e. Z ( F ` x ) e. S <-> A. c e. Z ( F ` c ) e. S )
23	5, 22	sylib 122	. . . . . . . 8 |- ( ph -> A. c e. Z ( F ` c ) e. S )
24	23	adantr 276	. . . . . . 7 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> A. c e. Z ( F ` c ) e. S )
25		peano2uz 9807	. . . . . . . . 9 |- ( x e. ( ZZ>= ` M ) -> ( x + 1 ) e. ( ZZ>= ` M ) )
26	25, 8	eleqtrrdi 2323	. . . . . . . 8 |- ( x e. ( ZZ>= ` M ) -> ( x + 1 ) e. Z )
27	13, 26	syl 14	. . . . . . 7 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x + 1 ) e. Z )
28	19, 24, 27	rspcdva 2913	. . . . . 6 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( F ` ( x + 1 ) ) e. S )
29	17, 14, 28	caovcld 6171	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( y .+ ( F ` ( x + 1 ) ) ) e. S )
30		fvoveq1 6036	. . . . . . 7 |- ( z = x -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
31	30	oveq2d 6029	. . . . . 6 |- ( z = x -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( x + 1 ) ) ) )
32		oveq1 6020	. . . . . 6 |- ( w = y -> ( w .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( F ` ( x + 1 ) ) ) )
33		eqid 2229	. . . . . 6 |- ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) = ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) )
34	31, 32, 33	ovmpog 6151	. . . . 5 |- ( ( x e. ( ZZ>= ` M ) /\ y e. S /\ ( y .+ ( F ` ( x + 1 ) ) ) e. S ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) = ( y .+ ( F ` ( x + 1 ) ) ) )
35	13, 14, 29, 34	syl3anc 1271	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) = ( y .+ ( F ` ( x + 1 ) ) ) )
36	35, 29	eqeltrd 2306	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. S )
37		iseqvalcbv 10711	. . 3 |- frec ( ( s e. ( ZZ>= ` M ) , t e. _V |-> <. ( s + 1 ) , ( s ( u e. ( ZZ>= ` M ) , v e. S |-> ( v .+ ( F ` ( u + 1 ) ) ) ) t ) >. ) , <. 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 ) >. )
38	8	eleq2i 2296	. . . . 5 |- ( x e. Z <-> x e. ( ZZ>= ` M ) )
39	38, 4	sylan2br 288	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
40	1, 37, 39, 15	seq3val 10712	. . 3 |- ( ph -> seq M ( .+ , F ) = ran frec ( ( s e. ( ZZ>= ` M ) , t e. _V |-> <. ( s + 1 ) , ( s ( u e. ( ZZ>= ` M ) , v e. S |-> ( v .+ ( F ` ( u + 1 ) ) ) ) t ) >. ) , <. M , ( F ` M ) >. ) )
41	1, 10, 12, 36, 37, 40	frecuzrdgtclt 10673	. 2 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
42	8	a1i 9	. . 3 |- ( ph -> Z = ( ZZ>= ` M ) )
43	42	feq2d 5467	. 2 |- ( ph -> ( seq M ( .+ , F ) : Z --> S <-> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S ) )
44	41, 43	mpbird 167	1 |- ( ph -> seq M ( .+ , F ) : Z --> S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2800   C_ wss 3198   <.cop 3670   -->wf 5320   ` cfv 5324  (class class class)co 6013   e. cmpo 6015  freccfrec 6551   1c1 8023   + caddc 8025   ZZcz 9469   ZZ>=cuz 9745   seqcseq 10699

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-seqfrec 10700

This theorem is referenced by:  seq3p1  10717  seq3feq2  10728  seq3feq  10732  serf  10735  serfre  10736  seq3split  10740  seq3caopr2  10745  seq3f1olemqsumkj  10763  seq3homo  10779  seq3z  10780  seqfeq3  10781  seq3distr  10784  ser3ge0  10788  exp3vallem  10792  exp3val  10793  facnn  10979  fac0  10980  bcval5  11015  seq3coll  11096  seq3shft  11389  resqrexlemf  11558  prodf  12089  algrf  12607  pcmptcl  12905  nninfdclemf  13060  mulgval  13699  mulgfng  13701  mulgnnsubcl  13711  lgsval  15723  lgscllem  15726  lgsval4a  15741  lgsneg  15743  lgsdir  15754  lgsdilem2  15755  lgsdi  15756  lgsne0  15757