Theorem algrf 11977

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 5699	. . . . 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 2243	. . 3 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) e. S )
8		vex 2729	. . . . 5 |- x e. _V
9		vex 2729	. . . . 5 |- y e. _V
10	8, 9	algrflem 6197	. . . 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 5618	. . . . 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 2253	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
16	1, 2, 7, 15	seqf 10396	. 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 5330	. 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 1343   e. wcel 2136   {csn 3576   X. cxp 4602   o. ccom 4608   -->wf 5184   ` cfv 5188  (class class class)co 5842   1stc1st 6106   ZZcz 9191   ZZ>=cuz 9466   seqcseq 10380

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-seqfrec 10381

This theorem is referenced by:  algrp1  11978  alginv  11979  algcvg  11980  algcvga  11983  algfx  11984  eucalgcvga  11990  eucalg  11991