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Theorem eucialgcvga 10665
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 5845 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  NN0 )
31, 2syl5eqel 2169 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  N  e. 
NN0 )
4 eluznn0 8837 . . . . . 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 8804 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
7 eucialg.2 . . . . . . 7  |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) ,  ( NN0  X.  NN0 ) )
8 0zd 8514 . . . . . . 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 10662 . . . . . . . 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 8431 . . . . . . . . 9  |-  NN0  e.  _V
1413, 13xpex 4501 . . . . . . . 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 10652 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  R : NN0
--> ( NN0  X.  NN0 ) )
1716ffvelrnda 5355 . . . . 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 5251 . . . 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 5251 . . . . . . . 8  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( 2nd `  A ) )
2322, 1syl6eqr 2133 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  N )
2423fveq2d 5234 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A ) )  =  ( ZZ>= `  N
) )
2524eleq2d 2152 . . . . 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 5839 . . . . 5  |-  ( 2nd  |`  ( NN0  X.  NN0 ) ) : ( NN0  X.  NN0 ) --> NN0
2810eucalglt 10664 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  ( E `
 z ) )  =/=  0  ->  ( 2nd `  ( E `  z ) )  < 
( 2nd `  z
) ) )
2911ffvelrni 5354 . . . . . . . 8  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( E `
 z )  e.  ( NN0  X.  NN0 ) )
30 fvres 5251 . . . . . . . 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 2267 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  <->  ( 2nd `  ( E `  z )
)  =/=  0 ) )
33 fvres 5251 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  z )  =  ( 2nd `  z ) )
3431, 33breq12d 3818 . . . . . 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 2083 . . . . 5  |-  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A )
3711, 7, 27, 35, 36, 14ialgcvga 10658 . . . 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 2117 . 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 1285    e. wcel 1434    =/= wne 2249   _Vcvv 2610   ifcif 3368   {csn 3416   <.cop 3419   class class class wbr 3805    X. cxp 4389    |` cres 4393    o. ccom 4395   -->wf 4948   ` cfv 4952  (class class class)co 5564    |-> cmpt2 5566   1stc1st 5817   2ndc2nd 5818   0cc0 7113    < clt 7285   NN0cn0 8425   ZZ>=cuz 8770    mod cmo 9474    seqcseq 9591
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226  ax-arch 7227
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771  df-q 8856  df-rp 8886  df-fl 9422  df-mod 9475  df-iseq 9592
This theorem is referenced by:  eucialg  10666
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