Theorem algrf 12823

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 5929	. . . . 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 2315	. . 3 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) e. S )
8		vex 2824	. . . . 5 |- x e. _V
9		vex 2824	. . . . 5 |- y e. _V
10	8, 9	algrflem 6465	. . . 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 5841	. . . . 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 2325	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
16	1, 2, 7, 15	seqf 10901	. 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 5526	. 2 |- ( R : Z --> S <-> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) : Z --> S )
19	16, 18	sylibr 134	1 |- ( ph -> R : Z --> S )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3709   X. cxp 4772   o. ccom 4778   -->wf 5373   ` cfv 5377  (class class class)co 6085   1stc1st 6372   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:  algrp1  12824  alginv  12825  algcvg  12826  algcvga  12829  algfx  12830  eucalgcvga  12836  eucalg  12837