Theorem ialgrf 10427

Description: An algorithm is a step function F : S --> S on a state space S. An algorithm acts on an initial state A e. S by iteratively applying F to give A, ( F ` A ), ( F ` ( F ` A ) ) and so on. An algorithm is said to halt if a fixed point of F is reached after a finite number of iterations.

The algorithm iterator R : NN0 --> S "runs" the algorithm F so that ( R ` k ) is the state after k iterations of F on the initial state A.

Domain and codomain of the algorithm iterator R. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Hypotheses

Ref	Expression
algrf.1	|- Z = ( ZZ>= ` M )
algrf.2	|- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) , S )
algrf.3	|- ( ph -> M e. ZZ )
algrf.4	|- ( ph -> A e. S )
algrf.5	|- ( ph -> F : S --> S )
algrf.s	|- ( ph -> S e. V )

Assertion

Ref	Expression
ialgrf	|- ( ph -> R : Z --> S )

Proof of Theorem ialgrf

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		algrf.1	. . 3 |- Z = ( ZZ>= ` M )
2		algrf.s	. . 3 |- ( ph -> S e. V )
3		algrf.3	. . 3 |- ( ph -> M e. ZZ )
4	1	eleq2i 2145	. . . 4 |- ( x e. Z <-> x e. ( ZZ>= ` M ) )
5		algrf.4	. . . . 5 |- ( ph -> A e. S )
6	1, 5	ialgrlemconst 10425	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )
7	4, 6	sylan2b 281	. . 3 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) e. S )
8		algrf.5	. . . 4 |- ( ph -> F : S --> S )
9	8	ialgrlem1st 10424	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
10	1, 2, 3, 7, 9	iseqf 9444	. 2 |- ( ph -> seq M ( ( F o. 1st ) , ( Z X. { A } ) , S ) : Z --> S )
11		algrf.2	. . 3 |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) , S )
12	11	feq1i 5059	. 2 |- ( R : Z --> S <-> seq M ( ( F o. 1st ) , ( Z X. { A } ) , S ) : Z --> S )
13	10, 12	sylibr 132	1 |- ( ph -> R : Z --> S )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {csn 3398   X. cxp 4361   o. ccom 4367   -->wf 4918   ` cfv 4922   1stc1st 5785   ZZcz 8351   ZZ>=cuz 8619   seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  ialginv  10429  ialgcvg  10430  ialgcvga  10433  ialgfx  10434  eucialgcvga  10440  eucialg  10441