Theorem algrf 12047

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 5732	. . . . 5 |- ( ( A e. S /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) = A )
5	3, 4	sylan 283	. . . 4 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) = A )
6	3	adantr 276	. . . 4 |- ( ( ph /\ x e. Z ) -> A e. S )
7	5, 6	eqeltrd 2254	. . 3 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) e. S )
8		vex 2742	. . . . 5 |- x e. _V
9		vex 2742	. . . . 5 |- y e. _V
10	8, 9	algrflem 6232	. . . 4 |- ( x ( F o. 1st ) y ) = ( F ` x )
11		algrf.5	. . . . 5 |- ( ph -> F : S --> S )
12		simpl 109	. . . . 5 |- ( ( x e. S /\ y e. S ) -> x e. S )
13		ffvelcdm 5651	. . . . 5 |- ( ( F : S --> S /\ x e. S ) -> ( F ` x ) e. S )
14	11, 12, 13	syl2an 289	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` x ) e. S )
15	10, 14	eqeltrid 2264	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
16	1, 2, 7, 15	seqf 10463	. 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 5360	. 2 |- ( R : Z --> S <-> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) : Z --> S )
19	16, 18	sylibr 134	1 |- ( ph -> R : Z --> S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {csn 3594   X. cxp 4626   o. ccom 4632   -->wf 5214   ` cfv 5218  (class class class)co 5877   1stc1st 6141   ZZcz 9255   ZZ>=cuz 9530   seqcseq 10447

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-seqfrec 10448

This theorem is referenced by:  algrp1  12048  alginv  12049  algcvg  12050  algcvga  12053  algfx  12054  eucalgcvga  12060  eucalg  12061