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Theorem bezoutlem1 12717
Description: Lemma for bezout 12721. (Contributed by Mario Carneiro, 15-Mar-2014.)
Hypotheses
Ref Expression
bezout.1  |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y
) ) }
bezout.3  |-  ( ph  ->  A  e.  ZZ )
bezout.4  |-  ( ph  ->  B  e.  ZZ )
Assertion
Ref Expression
bezoutlem1  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  M ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    ph, x, y, z
Allowed substitution hints:    M( x, y, z)

Proof of Theorem bezoutlem1
StepHypRef Expression
1 bezout.3 . . . 4  |-  ( ph  ->  A  e.  ZZ )
2 fveq2 5525 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  z )  =  ( abs `  A
) )
3 oveq1 5865 . . . . . . 7  |-  ( z  =  A  ->  (
z  x.  x )  =  ( A  x.  x ) )
42, 3eqeq12d 2297 . . . . . 6  |-  ( z  =  A  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
54rexbidv 2564 . . . . 5  |-  ( z  =  A  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) ) )
6 zre 10028 . . . . . 6  |-  ( z  e.  ZZ  ->  z  e.  RR )
7 1z 10053 . . . . . . . . 9  |-  1  e.  ZZ
8 ax-1rid 8807 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
z  x.  1 )  =  z )
98eqcomd 2288 . . . . . . . . 9  |-  ( z  e.  RR  ->  z  =  ( z  x.  1 ) )
10 oveq2 5866 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
z  x.  x )  =  ( z  x.  1 ) )
1110eqeq2d 2294 . . . . . . . . . 10  |-  ( x  =  1  ->  (
z  =  ( z  x.  x )  <->  z  =  ( z  x.  1 ) ) )
1211rspcev 2884 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  z  =  ( z  x.  1 ) )  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
137, 9, 12sylancr 644 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
14 eqeq1 2289 . . . . . . . . 9  |-  ( ( abs `  z )  =  z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  z  =  ( z  x.  x
) ) )
1514rexbidv 2564 . . . . . . . 8  |-  ( ( abs `  z )  =  z  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  z  =  ( z  x.  x ) ) )
1613, 15syl5ibrcom 213 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
17 znegcl 10055 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
187, 17ax-mp 8 . . . . . . . . 9  |-  -u 1  e.  ZZ
19 recn 8827 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
2019mulm1d 9231 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  -u z )
21 neg1cn 9813 . . . . . . . . . . 11  |-  -u 1  e.  CC
22 mulcom 8823 . . . . . . . . . . 11  |-  ( (
-u 1  e.  CC  /\  z  e.  CC )  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1
) )
2321, 19, 22sylancr 644 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1 ) )
2420, 23eqtr3d 2317 . . . . . . . . 9  |-  ( z  e.  RR  ->  -u z  =  ( z  x.  -u 1 ) )
25 oveq2 5866 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  (
z  x.  x )  =  ( z  x.  -u 1 ) )
2625eqeq2d 2294 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  ( -u z  =  ( z  x.  x )  <->  -u z  =  ( z  x.  -u 1
) ) )
2726rspcev 2884 . . . . . . . . 9  |-  ( (
-u 1  e.  ZZ  /\  -u z  =  (
z  x.  -u 1
) )  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
2818, 24, 27sylancr 644 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
29 eqeq1 2289 . . . . . . . . 9  |-  ( ( abs `  z )  =  -u z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  -u z  =  ( z  x.  x
) ) )
3029rexbidv 2564 . . . . . . . 8  |-  ( ( abs `  z )  =  -u z  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) ) )
3128, 30syl5ibrcom 213 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  -u z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
32 absor 11785 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  \/  ( abs `  z
)  =  -u z
) )
3316, 31, 32mpjaod 370 . . . . . 6  |-  ( z  e.  RR  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
346, 33syl 15 . . . . 5  |-  ( z  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
355, 34vtoclga 2849 . . . 4  |-  ( A  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) )
361, 35syl 15 . . 3  |-  ( ph  ->  E. x  e.  ZZ  ( abs `  A )  =  ( A  x.  x ) )
37 bezout.4 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ZZ )
3837zcnd 10118 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
3938adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ )  ->  B  e.  CC )
4039mul01d 9011 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( B  x.  0 )  =  0 )
4140oveq2d 5874 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( ( A  x.  x )  +  0 ) )
421zcnd 10118 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
43 zcn 10029 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
44 mulcl 8821 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
4542, 43, 44syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( A  x.  x )  e.  CC )
4645addid1d 9012 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  0 )  =  ( A  x.  x
) )
4741, 46eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( A  x.  x
) )
4847eqeq2d 2294 . . . . 5  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
49 0z 10035 . . . . . 6  |-  0  e.  ZZ
50 oveq2 5866 . . . . . . . . 9  |-  ( y  =  0  ->  ( B  x.  y )  =  ( B  x.  0 ) )
5150oveq2d 5874 . . . . . . . 8  |-  ( y  =  0  ->  (
( A  x.  x
)  +  ( B  x.  y ) )  =  ( ( A  x.  x )  +  ( B  x.  0 ) ) )
5251eqeq2d 2294 . . . . . . 7  |-  ( y  =  0  ->  (
( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  0 ) ) ) )
5352rspcev 2884 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) ) )  ->  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y
) ) )
5449, 53mpan 651 . . . . 5  |-  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
5548, 54syl6bir 220 . . . 4  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( A  x.  x )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
5655reximdva 2655 . . 3  |-  ( ph  ->  ( E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
5736, 56mpd 14 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
58 nnabscl 11809 . . . 4  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  NN )
5958ex 423 . . 3  |-  ( A  e.  ZZ  ->  ( A  =/=  0  ->  ( abs `  A )  e.  NN ) )
601, 59syl 15 . 2  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  NN ) )
61 eqeq1 2289 . . . . 5  |-  ( z  =  ( abs `  A
)  ->  ( z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
62612rexbidv 2586 . . . 4  |-  ( z  =  ( abs `  A
)  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x
)  +  ( B  x.  y ) )  <->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
63 bezout.1 . . . 4  |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y
) ) }
6462, 63elrab2 2925 . . 3  |-  ( ( abs `  A )  e.  M  <->  ( ( abs `  A )  e.  NN  /\  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
6564simplbi2com 1364 . 2  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) )  ->  ( ( abs `  A )  e.  NN  ->  ( abs `  A )  e.  M
) )
6657, 60, 65sylsyld 52 1  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   NNcn 9746   ZZcz 10024   abscabs 11719
This theorem is referenced by:  bezoutlem2  12718  bezoutlem4  12720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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