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Theorem eucalginv 11744
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 11742 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
32fveq2d 5425 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  X
) )  =  (  gcd  `  if (
( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) ) )
4 1st2nd2 6073 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
54adantr 274 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
65fveq2d 5425 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
7 df-ov 5777 . . . . . . 7  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
86, 7syl6eqr 2190 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
98oveq2d 5790 . . . . 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 9080 . . . . . . 7  |-  ( ( 2nd `  X )  e.  NN  ->  ( 2nd `  X )  e.  ZZ )
1110adantl 275 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 2nd `  X )  e.  ZZ )
12 xp1st 6063 . . . . . . . . . 10  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
1312adantr 274 . . . . . . . . 9  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 1st `  X )  e. 
NN0 )
1413nn0zd 9178 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 1st `  X )  e.  ZZ )
15 zmodcl 10124 . . . . . . . 8  |-  ( ( ( 1st `  X
)  e.  ZZ  /\  ( 2nd `  X )  e.  NN )  -> 
( ( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
1614, 15sylancom 416 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
1716nn0zd 9178 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e.  ZZ )
18 gcdcom 11669 . . . . . 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 408 . . . . 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 11686 . . . . . 6  |-  ( ( ( 1st `  X
)  e.  ZZ  /\  ( 2nd `  X )  e.  NN )  -> 
( ( ( 1st `  X )  mod  ( 2nd `  X ) )  gcd  ( 2nd `  X
) )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
2114, 20sylancom 416 . . . . 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 2176 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 2nd `  X
)  gcd  (  mod  `  X ) )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
23 nnne0 8755 . . . . . . . . 9  |-  ( ( 2nd `  X )  e.  NN  ->  ( 2nd `  X )  =/=  0 )
2423adantl 275 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 2nd `  X )  =/=  0 )
2524neneqd 2329 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  -.  ( 2nd `  X )  =  0 )
2625iffalsed 3484 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )
2726fveq2d 5425 . . . . 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 5777 . . . . 5  |-  ( ( 2nd `  X )  gcd  (  mod  `  X
) )  =  (  gcd  `  <. ( 2nd `  X ) ,  (  mod  `  X ) >. )
2927, 28syl6eqr 2190 . . . 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 5425 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  X )  =  (  gcd  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
31 df-ov 5777 . . . . 5  |-  ( ( 1st `  X )  gcd  ( 2nd `  X
) )  =  (  gcd  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
3230, 31syl6eqr 2190 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  X )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
3322, 29, 323eqtr4d 2182 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
34 iftrue 3479 . . . . 5  |-  ( ( 2nd `  X )  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  X )
3534fveq2d 5425 . . . 4  |-  ( ( 2nd `  X )  =  0  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
3635adantl 275 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  =  0 )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
37 xp2nd 6064 . . . 4  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  X )  e.  NN0 )
38 elnn0 8986 . . . 4  |-  ( ( 2nd `  X )  e.  NN0  <->  ( ( 2nd `  X )  e.  NN  \/  ( 2nd `  X
)  =  0 ) )
3937, 38sylib 121 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  X )  e.  NN  \/  ( 2nd `  X )  =  0 ) )
4033, 36, 39mpjaodan 787 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )  =  (  gcd  `  X
) )
413, 40eqtrd 2172 1  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  X
) )  =  (  gcd  `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480    =/= wne 2308   ifcif 3474   <.cop 3530    X. cxp 4537   ` cfv 5123  (class class class)co 5774    e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   0cc0 7627   NNcn 8727   NN0cn0 8984   ZZcz 9061    mod cmo 10102    gcd cgcd 11642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-fz 9798  df-fzo 9927  df-fl 10050  df-mod 10103  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-dvds 11501  df-gcd 11643
This theorem is referenced by:  eucalg  11747
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