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Theorem eucalgcvga 13004
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 6316 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  NN0 )
31, 2syl5eqel 2471 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  N  e. 
NN0 )
4 eluznn0 10478 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N ) )  ->  K  e.  NN0 )
53, 4sylan 458 . . . . 5  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  NN0 )
6 nn0uz 10452 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
7 eucalg.2 . . . . . . 7  |-  R  =  seq  0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) )
8 0z 10225 . . . . . . . 8  |-  0  e.  ZZ
98a1i 11 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  0  e.  ZZ )
10 id 20 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  A  e.  ( NN0  X.  NN0 ) )
11 eucalgval.1 . . . . . . . . 9  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
1211eucalgf 13001 . . . . . . . 8  |-  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
1312a1i 11 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
)
146, 7, 9, 10, 13algrf 12991 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  R : NN0
--> ( NN0  X.  NN0 ) )
1514ffvelrnda 5809 . . . . 5  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  NN0 )  ->  ( R `  K )  e.  ( NN0  X.  NN0 ) )
165, 15syldan 457 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( R `  K )  e.  ( NN0  X.  NN0 )
)
17 fvres 5685 . . . 4  |-  ( ( R `  K )  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  K ) )  =  ( 2nd `  ( R `  K
) ) )
1816, 17syl 16 . . 3  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( R `  K )
)  =  ( 2nd `  ( R `  K
) ) )
19 simpl 444 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  A  e.  ( NN0  X.  NN0 )
)
20 fvres 5685 . . . . . . . 8  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( 2nd `  A ) )
2120, 1syl6eqr 2437 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  N )
2221fveq2d 5672 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A ) )  =  ( ZZ>= `  N
) )
2322eleq2d 2454 . . . . 5  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  (
( 2nd  |`  ( NN0 
X.  NN0 ) ) `  A ) )  <->  K  e.  ( ZZ>= `  N )
) )
2423biimpar 472 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  A
) ) )
25 f2ndres 6308 . . . . 5  |-  ( 2nd  |`  ( NN0  X.  NN0 ) ) : ( NN0  X.  NN0 ) --> NN0
2611eucalglt 13003 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  ( E `
 z ) )  =/=  0  ->  ( 2nd `  ( E `  z ) )  < 
( 2nd `  z
) ) )
2712ffvelrni 5808 . . . . . . . 8  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( E `
 z )  e.  ( NN0  X.  NN0 ) )
28 fvres 5685 . . . . . . . 8  |-  ( ( E `  z )  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( 2nd `  ( E `  z
) ) )
2927, 28syl 16 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( 2nd `  ( E `  z
) ) )
3029neeq1d 2563 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  <->  ( 2nd `  ( E `  z )
)  =/=  0 ) )
31 fvres 5685 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  z )  =  ( 2nd `  z ) )
3229, 31breq12d 4166 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  < 
( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 z )  <->  ( 2nd `  ( E `  z
) )  <  ( 2nd `  z ) ) )
3326, 30, 323imtr4d 260 . . . . 5  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( E `  z )
)  <  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  z
) ) )
34 eqid 2387 . . . . 5  |-  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A )
3512, 7, 25, 33, 34algcvga 12997 . . . 4  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  (
( 2nd  |`  ( NN0 
X.  NN0 ) ) `  A ) )  -> 
( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  K ) )  =  0 ) )
3619, 24, 35sylc 58 . . 3  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( R `  K )
)  =  0 )
3718, 36eqtr3d 2421 . 2  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 2nd `  ( R `  K
) )  =  0 )
3837ex 424 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 359    = wceq 1649    e. wcel 1717    =/= wne 2550   ifcif 3682   {csn 3757   <.cop 3760   class class class wbr 4153    X. cxp 4816    |` cres 4820    o. ccom 4822   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286   2ndc2nd 6287   0cc0 8923    < clt 9053   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420    mod cmo 11177    seq cseq 11250
This theorem is referenced by:  eucalg  13005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fl 11129  df-mod 11178  df-seq 11251
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