ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eucalgcvga Unicode version

Theorem eucalgcvga 11766
Description: Once Euclid's Algorithm halts after  N steps, the second element of the state remains 0 . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 29-May-2014.)
Hypotheses
Ref Expression
eucalgval.1  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
eucalg.2  |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) )
eucalgcvga.3  |-  N  =  ( 2nd `  A
)
Assertion
Ref Expression
eucalgcvga  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  N
)  ->  ( 2nd `  ( R `  K
) )  =  0 ) )
Distinct variable groups:    x, y, N   
x, A, y    x, R
Allowed substitution hints:    R( y)    E( x, y)    K( x, y)

Proof of Theorem eucalgcvga
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eucalgcvga.3 . . . . . . 7  |-  N  =  ( 2nd `  A
)
2 xp2nd 6068 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  NN0 )
31, 2eqeltrid 2227 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  N  e. 
NN0 )
4 eluznn0 9416 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N ) )  ->  K  e.  NN0 )
53, 4sylan 281 . . . . 5  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  NN0 )
6 nn0uz 9380 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
7 eucalg.2 . . . . . . 7  |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) )
8 0zd 9086 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  0  e.  ZZ )
9 id 19 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  A  e.  ( NN0  X.  NN0 ) )
10 eucalgval.1 . . . . . . . . 9  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
1110eucalgf 11763 . . . . . . . 8  |-  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
1211a1i 9 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
)
136, 7, 8, 9, 12algrf 11753 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  R : NN0
--> ( NN0  X.  NN0 ) )
1413ffvelrnda 5559 . . . . 5  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  NN0 )  ->  ( R `  K )  e.  ( NN0  X.  NN0 ) )
155, 14syldan 280 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( R `  K )  e.  ( NN0  X.  NN0 )
)
16 fvres 5449 . . . 4  |-  ( ( R `  K )  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  K ) )  =  ( 2nd `  ( R `  K
) ) )
1715, 16syl 14 . . 3  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( R `  K )
)  =  ( 2nd `  ( R `  K
) ) )
18 simpl 108 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  A  e.  ( NN0  X.  NN0 )
)
19 fvres 5449 . . . . . . . 8  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( 2nd `  A ) )
2019, 1eqtr4di 2191 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  N )
2120fveq2d 5429 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A ) )  =  ( ZZ>= `  N
) )
2221eleq2d 2210 . . . . 5  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  (
( 2nd  |`  ( NN0 
X.  NN0 ) ) `  A ) )  <->  K  e.  ( ZZ>= `  N )
) )
2322biimpar 295 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  A
) ) )
24 f2ndres 6062 . . . . 5  |-  ( 2nd  |`  ( NN0  X.  NN0 ) ) : ( NN0  X.  NN0 ) --> NN0
2510eucalglt 11765 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  ( E `
 z ) )  =/=  0  ->  ( 2nd `  ( E `  z ) )  < 
( 2nd `  z
) ) )
2611ffvelrni 5558 . . . . . . . 8  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( E `
 z )  e.  ( NN0  X.  NN0 ) )
27 fvres 5449 . . . . . . . 8  |-  ( ( E `  z )  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( 2nd `  ( E `  z
) ) )
2826, 27syl 14 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( 2nd `  ( E `  z
) ) )
2928neeq1d 2327 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  <->  ( 2nd `  ( E `  z )
)  =/=  0 ) )
30 fvres 5449 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  z )  =  ( 2nd `  z ) )
3128, 30breq12d 3946 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  < 
( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 z )  <->  ( 2nd `  ( E `  z
) )  <  ( 2nd `  z ) ) )
3225, 29, 313imtr4d 202 . . . . 5  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( E `  z )
)  <  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  z
) ) )
33 eqid 2140 . . . . 5  |-  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A )
3411, 7, 24, 32, 33algcvga 11759 . . . 4  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  (
( 2nd  |`  ( NN0 
X.  NN0 ) ) `  A ) )  -> 
( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  K ) )  =  0 ) )
3518, 23, 34sylc 62 . . 3  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( R `  K )
)  =  0 )
3617, 35eqtr3d 2175 . 2  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 2nd `  ( R `  K
) )  =  0 )
3736ex 114 1  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  N
)  ->  ( 2nd `  ( R `  K
) )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481    =/= wne 2309   ifcif 3475   {csn 3528   <.cop 3531   class class class wbr 3933    X. cxp 4541    |` cres 4545    o. ccom 4547   -->wf 5123   ` cfv 5127  (class class class)co 5778    e. cmpo 5780   1stc1st 6040   2ndc2nd 6041   0cc0 7640    < clt 7820   NN0cn0 8997   ZZ>=cuz 9346    mod cmo 10122    seqcseq 10245
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 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-mulrcl 7739  ax-addcom 7740  ax-mulcom 7741  ax-addass 7742  ax-mulass 7743  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-1rid 7747  ax-0id 7748  ax-rnegex 7749  ax-precex 7750  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-ltwlin 7753  ax-pre-lttrn 7754  ax-pre-apti 7755  ax-pre-ltadd 7756  ax-pre-mulgt0 7757  ax-pre-mulext 7758  ax-arch 7759
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  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-rmo 2425  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-ilim 4295  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-frec 6292  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825  df-le 7826  df-sub 7955  df-neg 7956  df-reap 8357  df-ap 8364  df-div 8453  df-inn 8741  df-n0 8998  df-z 9075  df-uz 9347  df-q 9435  df-rp 9467  df-fl 10070  df-mod 10123  df-seqfrec 10246
This theorem is referenced by:  eucalg  11767
  Copyright terms: Public domain W3C validator