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Theorem eucalglt 11650
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 11647 . . . . . . 7  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
32adantr 274 . . . . . 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 3449 . . . . . . . . . . . . 13  |-  ( ( 2nd `  X )  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  X )
65eqeq2d 2129 . . . . . . . . . . . 12  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  <->  ( E `  X )  =  X ) )
7 fveq2 5389 . . . . . . . . . . . 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 2127 . . . . . . . . . . 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 2327 . . . . . . . 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 3454 . . . . . 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 2150 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( E `  X )  =  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
1615fveq2d 5393 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
17 xp2nd 6032 . . . . . 6  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  X )  e.  NN0 )
1817adantr 274 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e. 
NN0 )
19 1st2nd2 6041 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
2019adantr 274 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
2120fveq2d 5393 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
22 df-ov 5745 . . . . . . 7  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
2321, 22syl6eqr 2168 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
24 xp1st 6031 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
2524adantr 274 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e. 
NN0 )
2625nn0zd 9129 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e.  ZZ )
2713neqned 2292 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  =/=  0 )
28 elnnne0 8949 . . . . . . . 8  |-  ( ( 2nd `  X )  e.  NN  <->  ( ( 2nd `  X )  e. 
NN0  /\  ( 2nd `  X )  =/=  0
) )
2918, 27, 28sylanbrc 413 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  NN )
3026, 29zmodcld 10073 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
3123, 30eqeltrd 2194 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  e. 
NN0 )
32 op2ndg 6017 . . . . 5  |-  ( ( ( 2nd `  X
)  e.  NN0  /\  (  mod  `  X )  e.  NN0 )  ->  ( 2nd `  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )  =  (  mod  `  X
) )
3318, 31, 32syl2anc 408 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )  =  (  mod  `  X
) )
3416, 33, 233eqtrd 2154 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
35 zq 9374 . . . . 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 9129 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  ZZ )
38 zq 9374 . . . . 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 8728 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  0  <  ( 2nd `  X
) )
41 modqlt 10061 . . . 4  |-  ( ( ( 1st `  X
)  e.  QQ  /\  ( 2nd `  X )  e.  QQ  /\  0  <  ( 2nd `  X
) )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4236, 39, 40, 41syl3anc 1201 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4334, 42eqbrtrd 3920 . 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 1316    e. wcel 1465    =/= wne 2285   ifcif 3444   <.cop 3500   class class class wbr 3899    X. cxp 4507   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005   0cc0 7588    < clt 7768   NNcn 8684   NN0cn0 8935   ZZcz 9012   QQcq 9367    mod cmo 10050
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-n0 8936  df-z 9013  df-q 9368  df-rp 9398  df-fl 9998  df-mod 10051
This theorem is referenced by:  eucalgcvga  11651
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