Theorem algrf 11762

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 } ) )
algrf.3	|- ( ph -> M e. ZZ )
algrf.4	|- ( ph -> A e. S )
algrf.5	|- ( ph -> F : S --> S )

Assertion

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

Proof of Theorem algrf

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.3	. . 3 |- ( ph -> M e. ZZ )
3		algrf.4	. . . . 5 |- ( ph -> A e. S )
4		fvconst2g 5642	. . . . 5 |- ( ( A e. S /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) = A )
5	3, 4	sylan 281	. . . 4 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) = A )
6	3	adantr 274	. . . 4 |- ( ( ph /\ x e. Z ) -> A e. S )
7	5, 6	eqeltrd 2217	. . 3 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) e. S )
8		vex 2692	. . . . 5 |- x e. _V
9		vex 2692	. . . . 5 |- y e. _V
10	8, 9	algrflem 6134	. . . 4 |- ( x ( F o. 1st ) y ) = ( F ` x )
11		algrf.5	. . . . 5 |- ( ph -> F : S --> S )
12		simpl 108	. . . . 5 |- ( ( x e. S /\ y e. S ) -> x e. S )
13		ffvelrn 5561	. . . . 5 |- ( ( F : S --> S /\ x e. S ) -> ( F ` x ) e. S )
14	11, 12, 13	syl2an 287	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` x ) e. S )
15	10, 14	eqeltrid 2227	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
16	1, 2, 7, 15	seqf 10265	. 2 |- ( ph -> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) : Z --> S )
17		algrf.2	. . 3 |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )
18	17	feq1i 5273	. 2 |- ( R : Z --> S <-> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) : Z --> S )
19	16, 18	sylibr 133	1 |- ( ph -> R : Z --> S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   {csn 3532   X. cxp 4545   o. ccom 4551   -->wf 5127   ` cfv 5131  (class class class)co 5782   1stc1st 6044   ZZcz 9078   ZZ>=cuz 9350   seqcseq 10249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250

This theorem is referenced by:  algrp1  11763  alginv  11764  algcvg  11765  algcvga  11768  algfx  11769  eucalgcvga  11775  eucalg  11776