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Theorem eucalglt 10914
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 10911 . . . . . . 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 3384 . . . . . . . . . . . . 13  |-  ( ( 2nd `  X )  =  0  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  X )
65eqeq2d 2096 . . . . . . . . . . . 12  |-  ( ( 2nd `  X )  =  0  ->  (
( E `  X
)  =  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  <->  ( E `  X )  =  X ) )
7 fveq2 5268 . . . . . . . . . . . 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 2094 . . . . . . . . . . 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 2293 . . . . . . . 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 3389 . . . . . 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 2117 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( E `  X )  =  <. ( 2nd `  X
) ,  (  mod  `  X ) >. )
1615fveq2d 5272 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( 2nd `  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
17 xp2nd 5894 . . . . . 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 5902 . . . . . . . . 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 5272 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
22 df-ov 5616 . . . . . . 7  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
2321, 22syl6eqr 2135 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
24 xp1st 5893 . . . . . . . . 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 8799 . . . . . . 7  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 1st `  X )  e.  ZZ )
2713neqned 2258 . . . . . . . 8  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  =/=  0 )
28 elnnne0 8620 . . . . . . . 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 9680 . . . . . 6  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  e. 
NN0 )
3123, 30eqeltrd 2161 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (  mod  `  X )  e. 
NN0 )
32 op2ndg 5879 . . . . 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 2121 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  ( E `  X ) )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
35 zq 9043 . . . . 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 8799 . . . . 5  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  ( 2nd `  X )  e.  ZZ )
38 zq 9043 . . . . 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 8400 . . . 4  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  0  <  ( 2nd `  X
) )
41 modqlt 9668 . . . 4  |-  ( ( ( 1st `  X
)  e.  QQ  /\  ( 2nd `  X )  e.  QQ  /\  0  <  ( 2nd `  X
) )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4236, 39, 40, 41syl3anc 1172 . . 3  |-  ( ( X  e.  ( NN0 
X.  NN0 )  /\  ( 2nd `  ( E `  X ) )  =/=  0 )  ->  (
( 1st `  X
)  mod  ( 2nd `  X ) )  < 
( 2nd `  X
) )
4334, 42eqbrtrd 3840 . 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 1287    e. wcel 1436    =/= wne 2251   ifcif 3379   <.cop 3434   class class class wbr 3820    X. cxp 4409   ` cfv 4981  (class class class)co 5613    |-> cmpt2 5615   1stc1st 5866   2ndc2nd 5867   0cc0 7294    < clt 7466   NNcn 8357   NN0cn0 8606   ZZcz 8683   QQcq 9036    mod cmo 9657
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-1cn 7382  ax-1re 7383  ax-icn 7384  ax-addcl 7385  ax-addrcl 7386  ax-mulcl 7387  ax-mulrcl 7388  ax-addcom 7389  ax-mulcom 7390  ax-addass 7391  ax-mulass 7392  ax-distr 7393  ax-i2m1 7394  ax-0lt1 7395  ax-1rid 7396  ax-0id 7397  ax-rnegex 7398  ax-precex 7399  ax-cnre 7400  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403  ax-pre-apti 7404  ax-pre-ltadd 7405  ax-pre-mulgt0 7406  ax-pre-mulext 7407  ax-arch 7408
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-po 4097  df-iso 4098  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472  df-sub 7599  df-neg 7600  df-reap 7993  df-ap 8000  df-div 8079  df-inn 8358  df-n0 8607  df-z 8684  df-q 9037  df-rp 9067  df-fl 9605  df-mod 9658
This theorem is referenced by:  eucialgcvga  10915
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