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Theorem bezoutlem1 12993
Description: Lemma for bezout 12997. (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 5687 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  z )  =  ( abs `  A
) )
3 oveq1 6047 . . . . . . 7  |-  ( z  =  A  ->  (
z  x.  x )  =  ( A  x.  x ) )
42, 3eqeq12d 2418 . . . . . 6  |-  ( z  =  A  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
54rexbidv 2687 . . . . 5  |-  ( z  =  A  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) ) )
6 zre 10242 . . . . . 6  |-  ( z  e.  ZZ  ->  z  e.  RR )
7 1z 10267 . . . . . . . . 9  |-  1  e.  ZZ
8 ax-1rid 9016 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
z  x.  1 )  =  z )
98eqcomd 2409 . . . . . . . . 9  |-  ( z  e.  RR  ->  z  =  ( z  x.  1 ) )
10 oveq2 6048 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
z  x.  x )  =  ( z  x.  1 ) )
1110eqeq2d 2415 . . . . . . . . . 10  |-  ( x  =  1  ->  (
z  =  ( z  x.  x )  <->  z  =  ( z  x.  1 ) ) )
1211rspcev 3012 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  z  =  ( z  x.  1 ) )  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
137, 9, 12sylancr 645 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
14 eqeq1 2410 . . . . . . . . 9  |-  ( ( abs `  z )  =  z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  z  =  ( z  x.  x
) ) )
1514rexbidv 2687 . . . . . . . 8  |-  ( ( abs `  z )  =  z  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  z  =  ( z  x.  x ) ) )
1613, 15syl5ibrcom 214 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
17 znegcl 10269 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
187, 17ax-mp 8 . . . . . . . . 9  |-  -u 1  e.  ZZ
19 recn 9036 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
2019mulm1d 9441 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  -u z )
21 neg1cn 10023 . . . . . . . . . . 11  |-  -u 1  e.  CC
22 mulcom 9032 . . . . . . . . . . 11  |-  ( (
-u 1  e.  CC  /\  z  e.  CC )  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1
) )
2321, 19, 22sylancr 645 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1 ) )
2420, 23eqtr3d 2438 . . . . . . . . 9  |-  ( z  e.  RR  ->  -u z  =  ( z  x.  -u 1 ) )
25 oveq2 6048 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  (
z  x.  x )  =  ( z  x.  -u 1 ) )
2625eqeq2d 2415 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  ( -u z  =  ( z  x.  x )  <->  -u z  =  ( z  x.  -u 1
) ) )
2726rspcev 3012 . . . . . . . . 9  |-  ( (
-u 1  e.  ZZ  /\  -u z  =  (
z  x.  -u 1
) )  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
2818, 24, 27sylancr 645 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
29 eqeq1 2410 . . . . . . . . 9  |-  ( ( abs `  z )  =  -u z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  -u z  =  ( z  x.  x
) ) )
3029rexbidv 2687 . . . . . . . 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 214 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  -u z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
32 absor 12060 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  \/  ( abs `  z
)  =  -u z
) )
3316, 31, 32mpjaod 371 . . . . . 6  |-  ( z  e.  RR  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
346, 33syl 16 . . . . 5  |-  ( z  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
355, 34vtoclga 2977 . . . 4  |-  ( A  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) )
361, 35syl 16 . . 3  |-  ( ph  ->  E. x  e.  ZZ  ( abs `  A )  =  ( A  x.  x ) )
37 bezout.4 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ZZ )
3837zcnd 10332 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
3938adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ )  ->  B  e.  CC )
4039mul01d 9221 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( B  x.  0 )  =  0 )
4140oveq2d 6056 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( ( A  x.  x )  +  0 ) )
421zcnd 10332 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
43 zcn 10243 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
44 mulcl 9030 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
4542, 43, 44syl2an 464 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( A  x.  x )  e.  CC )
4645addid1d 9222 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  0 )  =  ( A  x.  x
) )
4741, 46eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( A  x.  x
) )
4847eqeq2d 2415 . . . . 5  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
49 0z 10249 . . . . . 6  |-  0  e.  ZZ
50 oveq2 6048 . . . . . . . . 9  |-  ( y  =  0  ->  ( B  x.  y )  =  ( B  x.  0 ) )
5150oveq2d 6056 . . . . . . . 8  |-  ( y  =  0  ->  (
( A  x.  x
)  +  ( B  x.  y ) )  =  ( ( A  x.  x )  +  ( B  x.  0 ) ) )
5251eqeq2d 2415 . . . . . . 7  |-  ( y  =  0  ->  (
( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  0 ) ) ) )
5352rspcev 3012 . . . . . 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 652 . . . . 5  |-  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
5548, 54syl6bir 221 . . . 4  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( A  x.  x )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
5655reximdva 2778 . . 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 15 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
58 nnabscl 12084 . . . 4  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  NN )
5958ex 424 . . 3  |-  ( A  e.  ZZ  ->  ( A  =/=  0  ->  ( abs `  A )  e.  NN ) )
601, 59syl 16 . 2  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  NN ) )
61 eqeq1 2410 . . . . 5  |-  ( z  =  ( abs `  A
)  ->  ( z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
62612rexbidv 2709 . . . 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 3054 . . 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 1380 . 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 54 1  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   {crab 2670   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   -ucneg 9248   NNcn 9956   ZZcz 10238   abscabs 11994
This theorem is referenced by:  bezoutlem2  12994  bezoutlem4  12996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996
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