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Theorem eucalglt 11531
Description: The second member of the state decreases with each iteration of the step function  E for Euclid's Algorithm. (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
eucalglt  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  ( E `
 X ) )  =/=  0  ->  ( 2nd `  ( E `  X ) )  < 
( 2nd `  X
) ) )
Distinct variable group:    x, y, X
Allowed substitution hints:    E( x, y)

Proof of Theorem eucalglt
StepHypRef Expression
1 eucalgval.1 . . . . . . . 8  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
21eucalgval 11528 . . . . . . 7  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
32adantr 272 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( E `  X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
4 simpr 109 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =/=  0 )
5 iftrue 3426 . . . . . . . . . . . . 13  |-  ( ( 2nd `  X )  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  X )
65eqeq2d 2111 . . . . . . . . . . . 12  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  <->  ( E `  X )  =  X ) )
7 fveq2 5353 . . . . . . . . . . . 12  |-  ( ( E `  X )  =  X  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  X
) )
86, 7syl6bi 162 . . . . . . . . . . 11  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  X
) ) )
9 eqeq2 2109 . . . . . . . . . . 11  |-  ( ( 2nd `  X )  =  0  ->  (
( 2nd `  ( E `  X )
)  =  ( 2nd `  X )  <->  ( 2nd `  ( E `  X
) )  =  0 ) )
108, 9sylibd 148 . . . . . . . . . 10  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  ->  ( 2nd `  ( E `  X ) )  =  0 ) )
113, 10syl5com 29 . . . . . . . . 9  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 2nd `  X
)  =  0  -> 
( 2nd `  ( E `  X )
)  =  0 ) )
1211necon3ad 2309 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 2nd `  ( E `  X )
)  =/=  0  ->  -.  ( 2nd `  X
)  =  0 ) )
134, 12mpd 13 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  -.  ( 2nd `  X )  =  0 )
1413iffalsed 3431 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )
153, 14eqtrd 2132 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( E `  X )  =  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
1615fveq2d 5357 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
17 xp2nd 5995 . . . . . 6  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  X )  e.  NN0 )
1817adantr 272 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e. 
NN0 )
19 1st2nd2 6003 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
2019adantr 272 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
2120fveq2d 5357 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
22 df-ov 5709 . . . . . . 7  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
2321, 22syl6eqr 2150 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
24 xp1st 5994 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
2524adantr 272 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e. 
NN0 )
2625nn0zd 9023 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e.  ZZ )
2713neqned 2274 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  =/=  0 )
28 elnnne0 8843 . . . . . . . 8  |-  ( ( 2nd `  X )  e.  NN  <->  ( ( 2nd `  X )  e. 
NN0  /\  ( 2nd `  X )  =/=  0
) )
2918, 27, 28sylanbrc 411 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  NN )
3026, 29zmodcld 9959 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
3123, 30eqeltrd 2176 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  e. 
NN0 )
32 op2ndg 5980 . . . . 5  |-  ( ( ( 2nd `  X
)  e.  NN0  /\  (  mod  `  X )  e.  NN0 )  ->  ( 2nd `  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )  =  (  mod  `  X
) )
3318, 31, 32syl2anc 406 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )  =  (  mod  `  X
) )
3416, 33, 233eqtrd 2136 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
35 zq 9268 . . . . 5  |-  ( ( 1st `  X )  e.  ZZ  ->  ( 1st `  X )  e.  QQ )
3626, 35syl 14 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e.  QQ )
3718nn0zd 9023 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  ZZ )
38 zq 9268 . . . . 5  |-  ( ( 2nd `  X )  e.  ZZ  ->  ( 2nd `  X )  e.  QQ )
3937, 38syl 14 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  QQ )
4029nngt0d 8622 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  0  <  ( 2nd `  X
) )
41 modqlt 9947 . . . 4  |-  ( ( ( 1st `  X
)  e.  QQ  /\  ( 2nd `  X )  e.  QQ  /\  0  <  ( 2nd `  X
) )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4236, 39, 40, 41syl3anc 1184 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4334, 42eqbrtrd 3895 . 2  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  < 
( 2nd `  X
) )
4443ex 114 1  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  ( E `
 X ) )  =/=  0  ->  ( 2nd `  ( E `  X ) )  < 
( 2nd `  X
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1299    e. wcel 1448    =/= wne 2267   ifcif 3421   <.cop 3477   class class class wbr 3875    X. cxp 4475   ` cfv 5059  (class class class)co 5706    e. cmpo 5708   1stc1st 5967   2ndc2nd 5968   0cc0 7500    < clt 7672   NNcn 8578   NN0cn0 8829   ZZcz 8906   QQcq 9261    mod cmo 9936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-n0 8830  df-z 8907  df-q 9262  df-rp 9292  df-fl 9884  df-mod 9937
This theorem is referenced by:  eucalgcvga  11532
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