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Theorem eucalginv 14542
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 14540 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
32fveq2d 5885 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  X
) )  =  (  gcd  `  if (
( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) ) )
4 1st2nd2 6844 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
54adantr 466 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
65fveq2d 5885 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
7 df-ov 6308 . . . . . . 7  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
86, 7syl6eqr 2481 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
98oveq2d 6321 . . . . 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 10966 . . . . . . 7  |-  ( ( 2nd `  X )  e.  NN  ->  ( 2nd `  X )  e.  ZZ )
1110adantl 467 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 2nd `  X )  e.  ZZ )
12 xp1st 6837 . . . . . . . . . 10  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
1312adantr 466 . . . . . . . . 9  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 1st `  X )  e. 
NN0 )
1413nn0zd 11045 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 1st `  X )  e.  ZZ )
15 zmodcl 12122 . . . . . . . 8  |-  ( ( ( 1st `  X
)  e.  ZZ  /\  ( 2nd `  X )  e.  NN )  -> 
( ( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
1614, 15sylancom 671 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
1716nn0zd 11045 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e.  ZZ )
18 gcdcom 14483 . . . . . 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 665 . . . . 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 14499 . . . . . 6  |-  ( ( ( 1st `  X
)  e.  ZZ  /\  ( 2nd `  X )  e.  NN )  -> 
( ( ( 1st `  X )  mod  ( 2nd `  X ) )  gcd  ( 2nd `  X
) )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
2114, 20sylancom 671 . . . . 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 2467 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (
( 2nd `  X
)  gcd  (  mod  `  X ) )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
23 nnne0 10649 . . . . . . . . 9  |-  ( ( 2nd `  X )  e.  NN  ->  ( 2nd `  X )  =/=  0 )
2423adantl 467 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  ( 2nd `  X )  =/=  0 )
2524neneqd 2621 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  -.  ( 2nd `  X )  =  0 )
2625iffalsed 3922 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )
2726fveq2d 5885 . . . . 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 6308 . . . . 5  |-  ( ( 2nd `  X )  gcd  (  mod  `  X
) )  =  (  gcd  `  <. ( 2nd `  X ) ,  (  mod  `  X ) >. )
2927, 28syl6eqr 2481 . . . 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 5885 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  X )  =  (  gcd  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
31 df-ov 6308 . . . . 5  |-  ( ( 1st `  X )  gcd  ( 2nd `  X
) )  =  (  gcd  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
3230, 31syl6eqr 2481 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  X )  =  ( ( 1st `  X
)  gcd  ( 2nd `  X ) ) )
3322, 29, 323eqtr4d 2473 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  e.  NN )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
34 iftrue 3917 . . . . 5  |-  ( ( 2nd `  X )  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  X )
3534fveq2d 5885 . . . 4  |-  ( ( 2nd `  X )  =  0  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
3635adantl 467 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  X )  =  0 )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
)  =  (  gcd  `  X ) )
37 xp2nd 6838 . . . 4  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  X )  e.  NN0 )
38 elnn0 10878 . . . 4  |-  ( ( 2nd `  X )  e.  NN0  <->  ( ( 2nd `  X )  e.  NN  \/  ( 2nd `  X
)  =  0 ) )
3937, 38sylib 199 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  X )  e.  NN  \/  ( 2nd `  X )  =  0 ) )
4033, 36, 39mpjaodan 793 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )  =  (  gcd  `  X
) )
413, 40eqtrd 2463 1  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  X
) )  =  (  gcd  `  X )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   ifcif 3911   <.cop 4004    X. cxp 4851   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   0cc0 9546   NNcn 10616   NN0cn0 10876   ZZcz 10944    mod cmo 12102    gcd cgcd 14467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fl 12034  df-mod 12103  df-seq 12220  df-exp 12279  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14305  df-gcd 14468
This theorem is referenced by:  eucalg  14545
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