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Theorem eucalglt 13081
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 . . . . . . . . 9  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
21eucalgval 13078 . . . . . . . 8  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
32adantr 453 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( E `  X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
4 simpr 449 . . . . . . . . 9  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =/=  0 )
5 iftrue 3747 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  X )  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  X )
65eqeq2d 2449 . . . . . . . . . . . . 13  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  <->  ( E `  X )  =  X ) )
7 fveq2 5731 . . . . . . . . . . . . 13  |-  ( ( E `  X )  =  X  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  X
) )
86, 7syl6bi 221 . . . . . . . . . . . 12  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  X
) ) )
9 eqeq2 2447 . . . . . . . . . . . 12  |-  ( ( 2nd `  X )  =  0  ->  (
( 2nd `  ( E `  X )
)  =  ( 2nd `  X )  <->  ( 2nd `  ( E `  X
) )  =  0 ) )
108, 9sylibd 207 . . . . . . . . . . 11  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  ->  ( 2nd `  ( E `  X ) )  =  0 ) )
113, 10syl5com 29 . . . . . . . . . 10  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 2nd `  X
)  =  0  -> 
( 2nd `  ( E `  X )
)  =  0 ) )
1211necon3ad 2639 . . . . . . . . 9  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 2nd `  ( E `  X )
)  =/=  0  ->  -.  ( 2nd `  X
)  =  0 ) )
134, 12mpd 15 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  -.  ( 2nd `  X )  =  0 )
14 iffalse 3748 . . . . . . . 8  |-  ( -.  ( 2nd `  X
)  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )
1513, 14syl 16 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )
163, 15eqtrd 2470 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( E `  X )  =  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
1716fveq2d 5735 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
18 fvex 5745 . . . . . 6  |-  ( 2nd `  X )  e.  _V
19 fvex 5745 . . . . . 6  |-  (  mod  `  X )  e.  _V
2018, 19op2nd 6359 . . . . 5  |-  ( 2nd `  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )  =  (  mod  `  X
)
2117, 20syl6eq 2486 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  (  mod  `  X
) )
22 1st2nd2 6389 . . . . . . 7  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
2322adantr 453 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
2423fveq2d 5735 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
25 df-ov 6087 . . . . 5  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
2624, 25syl6eqr 2488 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
2721, 26eqtrd 2470 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
28 xp1st 6379 . . . . . 6  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
2928adantr 453 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e. 
NN0 )
3029nn0red 10280 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e.  RR )
31 xp2nd 6380 . . . . . . . . 9  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  X )  e.  NN0 )
3231adantr 453 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e. 
NN0 )
33 elnn0 10228 . . . . . . . 8  |-  ( ( 2nd `  X )  e.  NN0  <->  ( ( 2nd `  X )  e.  NN  \/  ( 2nd `  X
)  =  0 ) )
3432, 33sylib 190 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 2nd `  X
)  e.  NN  \/  ( 2nd `  X )  =  0 ) )
3534ord 368 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( -.  ( 2nd `  X
)  e.  NN  ->  ( 2nd `  X )  =  0 ) )
3613, 35mt3d 120 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  NN )
3736nnrpd 10652 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  RR+ )
38 modlt 11263 . . . 4  |-  ( ( ( 1st `  X
)  e.  RR  /\  ( 2nd `  X )  e.  RR+ )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
3930, 37, 38syl2anc 644 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4027, 39eqbrtrd 4235 . 2  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  < 
( 2nd `  X
) )
4140ex 425 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    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ifcif 3741   <.cop 3819   class class class wbr 4215    X. cxp 4879   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   RRcr 8994   0cc0 8995    < clt 9125   NNcn 10005   NN0cn0 10226   RR+crp 10617    mod cmo 11255
This theorem is referenced by:  eucalgcvga  13082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fl 11207  df-mod 11256
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