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Theorem eucalgcvga 12756
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 6150 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  NN0 )
31, 2syl5eqel 2367 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  N  e. 
NN0 )
4 eluznn0 10288 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N ) )  ->  K  e.  NN0 )
53, 4sylan 457 . . . . 5  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  NN0 )
6 nn0uz 10262 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
7 eucalg.2 . . . . . . 7  |-  R  =  seq  0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) )
8 0z 10035 . . . . . . . 8  |-  0  e.  ZZ
98a1i 10 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  0  e.  ZZ )
10 id 19 . . . . . . 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 12753 . . . . . . . 8  |-  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
1312a1i 10 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
)
146, 7, 9, 10, 13algrf 12743 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  R : NN0
--> ( NN0  X.  NN0 ) )
15 ffvelrn 5663 . . . . . 6  |-  ( ( R : NN0 --> ( NN0 
X.  NN0 )  /\  K  e.  NN0 )  ->  ( R `  K )  e.  ( NN0  X.  NN0 ) )
1614, 15sylan 457 . . . . 5  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  NN0 )  ->  ( R `  K )  e.  ( NN0  X.  NN0 ) )
175, 16syldan 456 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( R `  K )  e.  ( NN0  X.  NN0 )
)
18 fvres 5542 . . . 4  |-  ( ( R `  K )  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  K ) )  =  ( 2nd `  ( R `  K
) ) )
1917, 18syl 15 . . 3  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( R `  K )
)  =  ( 2nd `  ( R `  K
) ) )
20 simpl 443 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  A  e.  ( NN0  X.  NN0 )
)
21 fvres 5542 . . . . . . . 8  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( 2nd `  A ) )
2221, 1syl6eqr 2333 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  N )
2322fveq2d 5529 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A ) )  =  ( ZZ>= `  N
) )
2423eleq2d 2350 . . . . 5  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  (
( 2nd  |`  ( NN0 
X.  NN0 ) ) `  A ) )  <->  K  e.  ( ZZ>= `  N )
) )
2524biimpar 471 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ( ZZ>= `  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  A
) ) )
26 f2ndres 6142 . . . . 5  |-  ( 2nd  |`  ( NN0  X.  NN0 ) ) : ( NN0  X.  NN0 ) --> NN0
2711eucalglt 12755 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  ( E `
 z ) )  =/=  0  ->  ( 2nd `  ( E `  z ) )  < 
( 2nd `  z
) ) )
2812ffvelrni 5664 . . . . . . . 8  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( E `
 z )  e.  ( NN0  X.  NN0 ) )
29 fvres 5542 . . . . . . . 8  |-  ( ( E `  z )  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( 2nd `  ( E `  z
) ) )
3028, 29syl 15 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( 2nd `  ( E `  z
) ) )
3130neeq1d 2459 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  <->  ( 2nd `  ( E `  z )
)  =/=  0 ) )
32 fvres 5542 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  z )  =  ( 2nd `  z ) )
3330, 32breq12d 4036 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  < 
( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 z )  <->  ( 2nd `  ( E `  z
) )  <  ( 2nd `  z ) ) )
3427, 31, 333imtr4d 259 . . . . 5  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( E `  z ) )  =/=  0  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( E `  z )
)  <  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  z
) ) )
35 eqid 2283 . . . . 5  |-  ( ( 2nd  |`  ( NN0  X. 
NN0 ) ) `  A )  =  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 A )
3612, 7, 26, 34, 35algcvga 12749 . . . 4  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  (
( 2nd  |`  ( NN0 
X.  NN0 ) ) `  A ) )  -> 
( ( 2nd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  K ) )  =  0 ) )
3720, 25, 36sylc 56 . . 3  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( 2nd  |`  ( NN0  X.  NN0 ) ) `  ( R `  K )
)  =  0 )
3819, 37eqtr3d 2317 . 2  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 2nd `  ( R `  K
) )  =  0 )
3938ex 423 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ifcif 3565   {csn 3640   <.cop 3643   class class class wbr 4023    X. cxp 4687    |` cres 4691    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   0cc0 8737    < clt 8867   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    mod cmo 10973    seq cseq 11046
This theorem is referenced by:  eucalg  12757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047
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