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Theorem eucalglt 10819
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 10816 . . . . . . 7  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
32adantr 270 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( E `  X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
4 simpr 108 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =/=  0 )
5 iftrue 3378 . . . . . . . . . . . . 13  |-  ( ( 2nd `  X )  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  X )
65eqeq2d 2094 . . . . . . . . . . . 12  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  <->  ( E `  X )  =  X ) )
7 fveq2 5253 . . . . . . . . . . . 12  |-  ( ( E `  X )  =  X  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  X
) )
86, 7syl6bi 161 . . . . . . . . . . 11  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  X
) ) )
9 eqeq2 2092 . . . . . . . . . . 11  |-  ( ( 2nd `  X )  =  0  ->  (
( 2nd `  ( E `  X )
)  =  ( 2nd `  X )  <->  ( 2nd `  ( E `  X
) )  =  0 ) )
108, 9sylibd 147 . . . . . . . . . 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 2291 . . . . . . . 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 3383 . . . . . 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 2115 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( E `  X )  =  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
1615fveq2d 5257 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
17 xp2nd 5872 . . . . . 6  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  X )  e.  NN0 )
1817adantr 270 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e. 
NN0 )
19 1st2nd2 5880 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
2019adantr 270 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
2120fveq2d 5257 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
22 df-ov 5594 . . . . . . 7  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
2321, 22syl6eqr 2133 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
24 xp1st 5871 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
2524adantr 270 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e. 
NN0 )
2625nn0zd 8762 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e.  ZZ )
2713neqned 2256 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  =/=  0 )
28 elnnne0 8579 . . . . . . . 8  |-  ( ( 2nd `  X )  e.  NN  <->  ( ( 2nd `  X )  e. 
NN0  /\  ( 2nd `  X )  =/=  0
) )
2918, 27, 28sylanbrc 408 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  NN )
3026, 29zmodcld 9641 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
3123, 30eqeltrd 2159 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  e. 
NN0 )
32 op2ndg 5857 . . . . 5  |-  ( ( ( 2nd `  X
)  e.  NN0  /\  (  mod  `  X )  e.  NN0 )  ->  ( 2nd `  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )  =  (  mod  `  X
) )
3318, 31, 32syl2anc 403 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )  =  (  mod  `  X
) )
3416, 33, 233eqtrd 2119 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
35 zq 9006 . . . . 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 8762 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  ZZ )
38 zq 9006 . . . . 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 8359 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  0  <  ( 2nd `  X
) )
41 modqlt 9629 . . . 4  |-  ( ( ( 1st `  X
)  e.  QQ  /\  ( 2nd `  X )  e.  QQ  /\  0  <  ( 2nd `  X
) )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4236, 39, 40, 41syl3anc 1170 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4334, 42eqbrtrd 3831 . 2  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  < 
( 2nd `  X
) )
4443ex 113 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 102    = wceq 1285    e. wcel 1434    =/= wne 2249   ifcif 3373   <.cop 3425   class class class wbr 3811    X. cxp 4399   ` cfv 4969  (class class class)co 5591    |-> cmpt2 5593   1stc1st 5844   2ndc2nd 5845   0cc0 7253    < clt 7425   NNcn 8316   NN0cn0 8565   ZZcz 8646   QQcq 8999    mod cmo 9618
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-mulrcl 7347  ax-addcom 7348  ax-mulcom 7349  ax-addass 7350  ax-mulass 7351  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-1rid 7355  ax-0id 7356  ax-rnegex 7357  ax-precex 7358  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-apti 7363  ax-pre-ltadd 7364  ax-pre-mulgt0 7365  ax-pre-mulext 7366  ax-arch 7367
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-po 4087  df-iso 4088  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-reap 7952  df-ap 7959  df-div 8038  df-inn 8317  df-n0 8566  df-z 8647  df-q 9000  df-rp 9030  df-fl 9566  df-mod 9619
This theorem is referenced by:  eucialgcvga  10820
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