Theorem seqf 10727

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 5639	. . . . 5 |- ( x = M -> ( F ` x ) = ( F ` M ) )
3	2	eleq1d 2300	. . . 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 2605	. . . 4 |- ( ph -> A. x e. Z ( F ` x ) e. S )
6		uzid 9770	. . . . . 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 2325	. . . 4 |- ( ph -> M e. Z )
10	3, 5, 9	rspcdva 2915	. . 3 |- ( ph -> ( F ` M ) e. S )
11		ssv 3249	. . . 4 |- S C_ _V
12	11	a1i 9	. . 3 |- ( ph -> S C_ _V )
13		simprl 531	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> x e. ( ZZ>= ` M ) )
14		simprr 533	. . . . 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 6175	. . . . . . 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 5639	. . . . . . . 8 |- ( c = ( x + 1 ) -> ( F ` c ) = ( F ` ( x + 1 ) ) )
19	18	eleq1d 2300	. . . . . . 7 |- ( c = ( x + 1 ) -> ( ( F ` c ) e. S <-> ( F ` ( x + 1 ) ) e. S ) )
20		fveq2 5639	. . . . . . . . . . 11 |- ( x = c -> ( F ` x ) = ( F ` c ) )
21	20	eleq1d 2300	. . . . . . . . . 10 |- ( x = c -> ( ( F ` x ) e. S <-> ( F ` c ) e. S ) )
22	21	cbvralv 2767	. . . . . . . . 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 9817	. . . . . . . . 9 |- ( x e. ( ZZ>= ` M ) -> ( x + 1 ) e. ( ZZ>= ` M ) )
26	25, 8	eleqtrrdi 2325	. . . . . . . 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 2915	. . . . . 6 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( F ` ( x + 1 ) ) e. S )
29	17, 14, 28	caovcld 6176	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( y .+ ( F ` ( x + 1 ) ) ) e. S )
30		fvoveq1 6041	. . . . . . 7 |- ( z = x -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
31	30	oveq2d 6034	. . . . . 6 |- ( z = x -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( x + 1 ) ) ) )
32		oveq1 6025	. . . . . 6 |- ( w = y -> ( w .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( F ` ( x + 1 ) ) ) )
33		eqid 2231	. . . . . 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 6156	. . . . 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 1273	. . . 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 2308	. . 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 10722	. . 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 2298	. . . . 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 10723	. . 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 10684	. 2 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
42	8	a1i 9	. . 3 |- ( ph -> Z = ( ZZ>= ` M ) )
43	42	feq2d 5470	. 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 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   <.cop 3672   -->wf 5322   ` cfv 5326  (class class class)co 6018   e. cmpo 6020  freccfrec 6556   1c1 8033   + caddc 8035   ZZcz 9479   ZZ>=cuz 9755   seqcseq 10710

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-seqfrec 10711

This theorem is referenced by:  seq3p1  10728  seq3feq2  10739  seq3feq  10743  serf  10746  serfre  10747  seq3split  10751  seq3caopr2  10756  seq3f1olemqsumkj  10774  seq3homo  10790  seq3z  10791  seqfeq3  10792  seq3distr  10795  ser3ge0  10799  exp3vallem  10803  exp3val  10804  facnn  10990  fac0  10991  bcval5  11026  seq3coll  11107  seq3shft  11416  resqrexlemf  11585  prodf  12117  algrf  12635  pcmptcl  12933  nninfdclemf  13088  mulgval  13727  mulgfng  13729  mulgnnsubcl  13739  lgsval  15752  lgscllem  15755  lgsval4a  15770  lgsneg  15772  lgsdir  15783  lgsdilem2  15784  lgsdi  15785  lgsne0  15786  depindlem1  16376