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Theorem bezoutlem1 12679
Description: Lemma for bezout 12683. (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 5458 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  z )  =  ( abs `  A
) )
3 oveq1 5799 . . . . . . 7  |-  ( z  =  A  ->  (
z  x.  x )  =  ( A  x.  x ) )
42, 3eqeq12d 2272 . . . . . 6  |-  ( z  =  A  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
54rexbidv 2539 . . . . 5  |-  ( z  =  A  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) ) )
6 zre 9995 . . . . . 6  |-  ( z  e.  ZZ  ->  z  e.  RR )
7 1z 10020 . . . . . . . . 9  |-  1  e.  ZZ
8 ax-1rid 8775 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
z  x.  1 )  =  z )
98eqcomd 2263 . . . . . . . . 9  |-  ( z  e.  RR  ->  z  =  ( z  x.  1 ) )
10 oveq2 5800 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
z  x.  x )  =  ( z  x.  1 ) )
1110eqeq2d 2269 . . . . . . . . . 10  |-  ( x  =  1  ->  (
z  =  ( z  x.  x )  <->  z  =  ( z  x.  1 ) ) )
1211rcla4ev 2859 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  z  =  ( z  x.  1 ) )  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
137, 9, 12sylancr 647 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
14 eqeq1 2264 . . . . . . . . 9  |-  ( ( abs `  z )  =  z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  z  =  ( z  x.  x
) ) )
1514rexbidv 2539 . . . . . . . 8  |-  ( ( abs `  z )  =  z  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  z  =  ( z  x.  x ) ) )
1613, 15syl5ibrcom 215 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
17 znegcl 10022 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
187, 17ax-mp 10 . . . . . . . . 9  |-  -u 1  e.  ZZ
19 recn 8795 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
2019mulm1d 9199 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  -u z )
21 neg1cn 9781 . . . . . . . . . . 11  |-  -u 1  e.  CC
22 mulcom 8791 . . . . . . . . . . 11  |-  ( (
-u 1  e.  CC  /\  z  e.  CC )  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1
) )
2321, 19, 22sylancr 647 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1 ) )
2420, 23eqtr3d 2292 . . . . . . . . 9  |-  ( z  e.  RR  ->  -u z  =  ( z  x.  -u 1 ) )
25 oveq2 5800 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  (
z  x.  x )  =  ( z  x.  -u 1 ) )
2625eqeq2d 2269 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  ( -u z  =  ( z  x.  x )  <->  -u z  =  ( z  x.  -u 1
) ) )
2726rcla4ev 2859 . . . . . . . . 9  |-  ( (
-u 1  e.  ZZ  /\  -u z  =  (
z  x.  -u 1
) )  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
2818, 24, 27sylancr 647 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
29 eqeq1 2264 . . . . . . . . 9  |-  ( ( abs `  z )  =  -u z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  -u z  =  ( z  x.  x
) ) )
3029rexbidv 2539 . . . . . . . 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 215 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  -u z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
32 absor 11750 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  \/  ( abs `  z
)  =  -u z
) )
3316, 31, 32mpjaod 372 . . . . . 6  |-  ( z  e.  RR  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
346, 33syl 17 . . . . 5  |-  ( z  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
355, 34vtoclga 2824 . . . 4  |-  ( A  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) )
361, 35syl 17 . . 3  |-  ( ph  ->  E. x  e.  ZZ  ( abs `  A )  =  ( A  x.  x ) )
37 bezout.4 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ZZ )
3837zcnd 10085 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
3938adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ )  ->  B  e.  CC )
4039mul01d 8979 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( B  x.  0 )  =  0 )
4140oveq2d 5808 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( ( A  x.  x )  +  0 ) )
421zcnd 10085 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
43 zcn 9996 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
44 mulcl 8789 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
4542, 43, 44syl2an 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( A  x.  x )  e.  CC )
4645addid1d 8980 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  0 )  =  ( A  x.  x
) )
4741, 46eqtrd 2290 . . . . . 6  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( A  x.  x
) )
4847eqeq2d 2269 . . . . 5  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
49 0z 10002 . . . . . 6  |-  0  e.  ZZ
50 oveq2 5800 . . . . . . . . 9  |-  ( y  =  0  ->  ( B  x.  y )  =  ( B  x.  0 ) )
5150oveq2d 5808 . . . . . . . 8  |-  ( y  =  0  ->  (
( A  x.  x
)  +  ( B  x.  y ) )  =  ( ( A  x.  x )  +  ( B  x.  0 ) ) )
5251eqeq2d 2269 . . . . . . 7  |-  ( y  =  0  ->  (
( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  0 ) ) ) )
5352rcla4ev 2859 . . . . . 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 654 . . . . 5  |-  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
5548, 54syl6bir 222 . . . 4  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( A  x.  x )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
5655reximdva 2630 . . 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 16 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
58 nnabscl 11774 . . . 4  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  NN )
5958ex 425 . . 3  |-  ( A  e.  ZZ  ->  ( A  =/=  0  ->  ( abs `  A )  e.  NN ) )
601, 59syl 17 . 2  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  NN ) )
61 eqeq1 2264 . . . . 5  |-  ( z  =  ( abs `  A
)  ->  ( z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
62612rexbidv 2561 . . . 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 2900 . . 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 1370 . 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 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   {crab 2522   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710   -ucneg 9006   NNcn 9714   ZZcz 9991   abscabs 11684
This theorem is referenced by:  bezoutlem2  12680  bezoutlem4  12682
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-seq 11013  df-exp 11071  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686
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