Theorem algcvgb 11720

Description: Two ways of expressing that C is a countdown function for algorithm F. The first is used in these theorems. The second states the condition more intuitively as a conjunction: if the countdown function's value is currently nonzero, it must decrease at the next step; if it has reached zero, it must remain zero at the next step. (Contributed by Paul Chapman, 31-Mar-2011.)

Hypotheses

Ref	Expression
algcvgb.1	|- F : S --> S
algcvgb.2	|- C : S --> NN0

Assertion

Ref	Expression
algcvgb	|- ( X e. S -> ( ( ( C ` ( F ` X ) ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) <-> ( ( ( C ` X ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) /\ ( ( C ` X ) = 0 -> ( C ` ( F ` X ) ) = 0 ) ) ) )

Proof of Theorem algcvgb

Step	Hyp	Ref	Expression
1		algcvgb.2	. . 3 |- C : S --> NN0
2	1	ffvelrni 5547	. 2 |- ( X e. S -> ( C ` X ) e. NN0 )
3		algcvgb.1	. . . 4 |- F : S --> S
4	3	ffvelrni 5547	. . 3 |- ( X e. S -> ( F ` X ) e. S )
5	1	ffvelrni 5547	. . 3 |- ( ( F ` X ) e. S -> ( C ` ( F ` X ) ) e. NN0 )
6	4, 5	syl 14	. 2 |- ( X e. S -> ( C ` ( F ` X ) ) e. NN0 )
7		algcvgblem 11719	. 2 |- ( ( ( C ` X ) e. NN0 /\ ( C ` ( F ` X ) ) e. NN0 ) -> ( ( ( C ` ( F ` X ) ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) <-> ( ( ( C ` X ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) /\ ( ( C ` X ) = 0 -> ( C ` ( F ` X ) ) = 0 ) ) ) )
8	2, 6, 7	syl2anc 408	1 |- ( X e. S -> ( ( ( C ` ( F ` X ) ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) <-> ( ( ( C ` X ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) /\ ( ( C ` X ) = 0 -> ( C ` ( F ` X ) ) = 0 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   =/= wne 2306   class class class wbr 3924   -->wf 5114   ` cfv 5118   0cc0 7613   < clt 7793   NN0cn0 8970

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048

This theorem is referenced by:  algcvga  11721