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

Theorem eucialgcvga 11120
Description: Once Euclid's Algorithm halts after  N steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.)
Hypotheses
Ref Expression
eucalgval.1  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
eucialg.2  |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) ,  ( NN0  X.  NN0 ) )
eucialgcvga.3  |-  N  =  ( 2nd `  A
)
Assertion
Ref Expression
eucialgcvga  |-  ( 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 eucialgcvga
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eucialgcvga.3 . . . . . . 7  |-  N  =  ( 2nd `  A
)
2 xp2nd 5919 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  NN0 )
31, 2syl5eqel 2174 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  N  e. 
NN0 )
4 eluznn0 9055 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N ) )  ->  K  e.  NN0 )
53, 4sylan 277 . . . . 5  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  NN0 )
6 nn0uz 9022 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
7 eucialg.2 . . . . . . 7  |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) ,  ( NN0  X.  NN0 ) )
8 0zd 8732 . . . . . . 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 11117 . . . . . . . 8  |-  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
1211a1i 9 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
)
13 nn0ex 8649 . . . . . . . . 9  |-  NN0  e.  _V
1413, 13xpex 4541 . . . . . . . 8  |-  ( NN0 
X.  NN0 )  e.  _V
1514a1i 9 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( NN0 
X.  NN0 )  e.  _V )
166, 7, 8, 9, 12, 15ialgrf 11107 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  R : NN0
--> ( NN0  X.  NN0 ) )
1716ffvelrnda 5418 . . . . 5  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  NN0 )  ->  ( R `  K )  e.  ( NN0  X.  NN0 ) )
185, 17syldan 276 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( R `  K )  e.  ( NN0  X.  NN0 )
)
19 fvres 5313 . . . 4  |-  ( ( R `  K )  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  K ) )  =  ( 2nd `  ( R `  K
) ) )
2018, 19syl 14 . . 3  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( R `  K )
)  =  ( 2nd `  ( R `  K
) ) )
21 simpl 107 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  A  e.  ( NN0  X.  NN0 )
)
22 fvres 5313 . . . . . . . 8  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( 2nd `  A ) )
2322, 1syl6eqr 2138 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  N )
2423fveq2d 5293 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A ) )  =  ( ZZ>= `  N
) )
2524eleq2d 2157 . . . . 5  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  (
( 2nd  |`  ( NN0 
X.  NN0 ) ) `  A ) )  <->  K  e.  ( ZZ>= `  N )
) )
2625biimpar 291 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  A
) ) )
27 f2ndres 5913 . . . . 5  |-  ( 2nd  |`  ( NN0  X.  NN0 ) ) : ( NN0  X.  NN0 ) --> NN0
2810eucalglt 11119 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  ( E `
 z ) )  =/=  0  ->  ( 2nd `  ( E `  z ) )  < 
( 2nd `  z
) ) )
2911ffvelrni 5417 . . . . . . . 8  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( E `
 z )  e.  ( NN0  X.  NN0 ) )
30 fvres 5313 . . . . . . . 8  |-  ( ( E `  z )  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( 2nd `  ( E `  z
) ) )
3129, 30syl 14 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( 2nd `  ( E `  z
) ) )
3231neeq1d 2273 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  <->  ( 2nd `  ( E `  z )
)  =/=  0 ) )
33 fvres 5313 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  z )  =  ( 2nd `  z ) )
3431, 33breq12d 3850 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  < 
( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 z )  <->  ( 2nd `  ( E `  z
) )  <  ( 2nd `  z ) ) )
3528, 32, 343imtr4d 201 . . . . 5  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( E `  z )
)  <  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  z
) ) )
36 eqid 2088 . . . . 5  |-  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A )
3711, 7, 27, 35, 36, 14ialgcvga 11113 . . . 4  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  (
( 2nd  |`  ( NN0 
X.  NN0 ) ) `  A ) )  -> 
( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  K ) )  =  0 ) )
3821, 26, 37sylc 61 . . 3  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( R `  K )
)  =  0 )
3920, 38eqtr3d 2122 . 2  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 2nd `  ( R `  K
) )  =  0 )
4039ex 113 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 102    = wceq 1289    e. wcel 1438    =/= wne 2255   _Vcvv 2619   ifcif 3389   {csn 3441   <.cop 3444   class class class wbr 3837    X. cxp 4426    |` cres 4430    o. ccom 4432   -->wf 4998   ` cfv 5002  (class class class)co 5634    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892   0cc0 7329    < clt 7501   NN0cn0 8643   ZZ>=cuz 8988    mod cmo 9694    seqcseq4 9816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fl 9642  df-mod 9695  df-iseq 9818
This theorem is referenced by:  eucialg  11121
  Copyright terms: Public domain W3C validator