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Theorem eucalginv 11131
Description: The invariant of the step function  E for Euclid's Algorithm is the  gcd operator applied to the state. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)
Hypothesis
Ref Expression
eucalgval.1  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
Assertion
Ref Expression
eucalginv  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  X
) )  =  (  gcd  `  X )
)
Distinct variable group:    x, y, X
Allowed substitution hints:    E( x, y)

Proof of Theorem eucalginv
StepHypRef Expression
1 eucalgval.1 . . . 4  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
21eucalgval 11129 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
32fveq2d 5293 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  X
) )  =  (  gcd  `  if (
( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) ) )
4 1st2nd2 5927 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
54adantr 270 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
65fveq2d 5293 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
7 df-ov 5637 . . . . . . 7  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
86, 7syl6eqr 2138 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
98oveq2d 5650 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 2nd `  X
)  gcd  (  mod  `  X ) )  =  ( ( 2nd `  X
)  gcd  ( ( 1st `  X )  mod  ( 2nd `  X
) ) ) )
10 nnz 8739 . . . . . . 7  |-  ( ( 2nd `  X )  e.  NN  ->  ( 2nd `  X )  e.  ZZ )
1110adantl 271 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 2nd `  X )  e.  ZZ )
12 xp1st 5918 . . . . . . . . . 10  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
1312adantr 270 . . . . . . . . 9  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 1st `  X )  e. 
NN0 )
1413nn0zd 8836 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 1st `  X )  e.  ZZ )
15 zmodcl 9716 . . . . . . . 8  |-  ( ( ( 1st `  X
)  e.  ZZ  /\  ( 2nd `  X )  e.  NN )  -> 
( ( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
1614, 15sylancom 411 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
1716nn0zd 8836 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e.  ZZ )
18 gcdcom 11058 . . . . . 6  |-  ( ( ( 2nd `  X
)  e.  ZZ  /\  ( ( 1st `  X
)  mod  ( 2nd `  X ) )  e.  ZZ )  ->  (
( 2nd `  X
)  gcd  ( ( 1st `  X )  mod  ( 2nd `  X
) ) )  =  ( ( ( 1st `  X )  mod  ( 2nd `  X ) )  gcd  ( 2nd `  X
) ) )
1911, 17, 18syl2anc 403 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 2nd `  X
)  gcd  ( ( 1st `  X )  mod  ( 2nd `  X
) ) )  =  ( ( ( 1st `  X )  mod  ( 2nd `  X ) )  gcd  ( 2nd `  X
) ) )
20 modgcd 11075 . . . . . 6  |-  ( ( ( 1st `  X
)  e.  ZZ  /\  ( 2nd `  X )  e.  NN )  -> 
( ( ( 1st `  X )  mod  ( 2nd `  X ) )  gcd  ( 2nd `  X
) )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
2114, 20sylancom 411 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( ( 1st `  X
)  mod  ( 2nd `  X ) )  gcd  ( 2nd `  X
) )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
229, 19, 213eqtrd 2124 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 2nd `  X
)  gcd  (  mod  `  X ) )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
23 nnne0 8422 . . . . . . . . 9  |-  ( ( 2nd `  X )  e.  NN  ->  ( 2nd `  X )  =/=  0 )
2423adantl 271 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 2nd `  X )  =/=  0 )
2524neneqd 2276 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  -.  ( 2nd `  X )  =  0 )
2625iffalsed 3399 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )
2726fveq2d 5293 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)
28 df-ov 5637 . . . . 5  |-  ( ( 2nd `  X )  gcd  (  mod  `  X
) )  =  (  gcd  `  <. ( 2nd `  X ) ,  (  mod  `  X ) >. )
2927, 28syl6eqr 2138 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  ( ( 2nd `  X )  gcd  (  mod  `  X
) ) )
305fveq2d 5293 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  X )  =  (  gcd  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
31 df-ov 5637 . . . . 5  |-  ( ( 1st `  X )  gcd  ( 2nd `  X
) )  =  (  gcd  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
3230, 31syl6eqr 2138 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  X )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
3322, 29, 323eqtr4d 2130 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
34 iftrue 3394 . . . . 5  |-  ( ( 2nd `  X )  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  X )
3534fveq2d 5293 . . . 4  |-  ( ( 2nd `  X )  =  0  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
3635adantl 271 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  =  0 )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
37 xp2nd 5919 . . . 4  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  X )  e.  NN0 )
38 elnn0 8645 . . . 4  |-  ( ( 2nd `  X )  e.  NN0  <->  ( ( 2nd `  X )  e.  NN  \/  ( 2nd `  X
)  =  0 ) )
3937, 38sylib 120 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  X )  e.  NN  \/  ( 2nd `  X )  =  0 ) )
4033, 36, 39mpjaodan 747 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )  =  (  gcd  `  X
) )
413, 40eqtrd 2120 1  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  X
) )  =  (  gcd  `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438    =/= wne 2255   ifcif 3389   <.cop 3444    X. cxp 4426   ` cfv 5002  (class class class)co 5634    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892   0cc0 7329   NNcn 8394   NN0cn0 8643   ZZcz 8720    mod cmo 9694    gcd cgcd 11031
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  ax-caucvg 7444
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-sup 6658  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-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10890  df-gcd 11032
This theorem is referenced by:  eucialg  11134
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