Theorem seqf 10901

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 5695	. . . . 5 |- ( x = M -> ( F ` x ) = ( F ` M ) )
3	2	eleq1d 2307	. . . 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 2623	. . . 4 |- ( ph -> A. x e. Z ( F ` x ) e. S )
6		uzid 9936	. . . . . 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 2332	. . . 4 |- ( ph -> M e. Z )
10	3, 5, 9	rspcdva 2934	. . 3 |- ( ph -> ( F ` M ) e. S )
11		ssv 3270	. . . 4 |- S C_ _V
12	11	a1i 9	. . 3 |- ( ph -> S C_ _V )
13		simprl 535	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> x e. ( ZZ>= ` M ) )
14		simprr 537	. . . . 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 6242	. . . . . . 7 |- ( ( ph /\ ( a e. S /\ b e. S ) ) -> ( a .+ b ) e. S )
17	16	adantlr 481	. . . . . 6 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( a e. S /\ b e. S ) ) -> ( a .+ b ) e. S )
18		fveq2 5695	. . . . . . . 8 |- ( c = ( x + 1 ) -> ( F ` c ) = ( F ` ( x + 1 ) ) )
19	18	eleq1d 2307	. . . . . . 7 |- ( c = ( x + 1 ) -> ( ( F ` c ) e. S <-> ( F ` ( x + 1 ) ) e. S ) )
20		fveq2 5695	. . . . . . . . . . 11 |- ( x = c -> ( F ` x ) = ( F ` c ) )
21	20	eleq1d 2307	. . . . . . . . . 10 |- ( x = c -> ( ( F ` x ) e. S <-> ( F ` c ) e. S ) )
22	21	cbvralv 2786	. . . . . . . . 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 9983	. . . . . . . . 9 |- ( x e. ( ZZ>= ` M ) -> ( x + 1 ) e. ( ZZ>= ` M ) )
26	25, 8	eleqtrrdi 2332	. . . . . . . 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 2934	. . . . . 6 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( F ` ( x + 1 ) ) e. S )
29	17, 14, 28	caovcld 6243	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( y .+ ( F ` ( x + 1 ) ) ) e. S )
30		fvoveq1 6108	. . . . . . 7 |- ( z = x -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
31	30	oveq2d 6101	. . . . . 6 |- ( z = x -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( x + 1 ) ) ) )
32		oveq1 6092	. . . . . 6 |- ( w = y -> ( w .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( F ` ( x + 1 ) ) ) )
33		eqid 2238	. . . . . 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 6223	. . . . 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 1278	. . . 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 2315	. . 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 10896	. . 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 2305	. . . . 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 10897	. . 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 10858	. 2 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
42	8	a1i 9	. . 3 |- ( ph -> Z = ( ZZ>= ` M ) )
43	42	feq2d 5521	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   <.cop 3712   -->wf 5373   ` cfv 5377  (class class class)co 6085   e. cmpo 6087  freccfrec 6661   1c1 8180   + caddc 8182   ZZcz 9644   ZZ>=cuz 9921   seqcseq 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885

This theorem is used by:  seq3p1  10902  seq3feq2  10913  seq3feq  10917  serf  10920  serfre  10921  seq3split  10925  seq3caopr2  10930  seq3f1olemqsumkj  10948  seq3homo  10964  seq3z  10965  seqfeq3  10966  seq3distr  10969  ser3ge0  10973  exp3vallem  10977  exp3val  10978  facnn  11165  fac0  11166  bcval5  11201  seq3coll  11294  seq3shft  11603  resqrexlemf  11773  prodf  12305  algrf  12823  pcmptcl  13121  nninfdclemf  13340  mulgval  13925  mulgfng  13927  mulgnnsubcl  13937  logfac  15995  lgsval  16123  lgscllem  16126  lgsval4a  16141  lgsneg  16143  lgsdir  16154  lgsdilem2  16155  lgsdi  16156  lgsne0  16157  depindlem1  16747