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Theorem aaitgo 27346
Description: The standard algebraic numbers  AA are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
aaitgo  |-  AA  =  (IntgOver `  QQ )

Proof of Theorem aaitgo
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabid 2886 . . 3  |-  ( a  e.  { a  e.  CC  |  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }  <->  ( a  e.  CC  /\  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
2 qsscn 10587 . . . . 5  |-  QQ  C_  CC
3 itgoval 27345 . . . . 5  |-  ( QQ  C_  CC  ->  (IntgOver `  QQ )  =  { a  e.  CC  |  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
42, 3ax-mp 8 . . . 4  |-  (IntgOver `  QQ )  =  { a  e.  CC  |  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }
54eleq2i 2502 . . 3  |-  ( a  e.  (IntgOver `  QQ ) 
<->  a  e.  { a  e.  CC  |  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
6 aacn 20236 . . . . 5  |-  ( a  e.  AA  ->  a  e.  CC )
7 mpaacl 27337 . . . . . 6  |-  ( a  e.  AA  ->  (minPolyAA `  a )  e.  (Poly `  QQ ) )
8 mpaaroot 27339 . . . . . 6  |-  ( a  e.  AA  ->  (
(minPolyAA `  a ) `  a )  =  0 )
9 mpaadgr 27338 . . . . . . . 8  |-  ( a  e.  AA  ->  (deg `  (minPolyAA `  a )
)  =  (degAA `  a
) )
109fveq2d 5734 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  ( (coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) ) )
11 mpaamn 27340 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) )  =  1 )
1210, 11eqtrd 2470 . . . . . 6  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 )
13 fveq1 5729 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( b `  a )  =  ( (minPolyAA `  a ) `  a ) )
1413eqeq1d 2446 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
b `  a )  =  0  <->  ( (minPolyAA `  a ) `  a
)  =  0 ) )
15 fveq2 5730 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (coeff `  b
)  =  (coeff `  (minPolyAA `  a ) ) )
16 fveq2 5730 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (deg `  b
)  =  (deg `  (minPolyAA `  a ) ) )
1715, 16fveq12d 5736 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( (coeff `  b ) `  (deg `  b ) )  =  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) ) )
1817eqeq1d 2446 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
(coeff `  b ) `  (deg `  b )
)  =  1  <->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 ) )
1914, 18anbi12d 693 . . . . . . 7  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 )  <->  ( ( (minPolyAA `  a ) `  a
)  =  0  /\  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) )  =  1 ) ) )
2019rspcev 3054 . . . . . 6  |-  ( ( (minPolyAA `  a )  e.  (Poly `  QQ )  /\  ( ( (minPolyAA `  a
) `  a )  =  0  /\  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 ) )  ->  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )
217, 8, 12, 20syl12anc 1183 . . . . 5  |-  ( a  e.  AA  ->  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )
226, 21jca 520 . . . 4  |-  ( a  e.  AA  ->  (
a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
23 simpl 445 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  e.  (Poly `  QQ )
)
24 coe0 20176 . . . . . . . . . . . . . . 15  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
2524fveq1i 5731 . . . . . . . . . . . . . 14  |-  ( (coeff `  0 p ) `
 (deg `  0 p ) )  =  ( ( NN0  X.  { 0 } ) `
 (deg `  0 p ) )
26 dgr0 20182 . . . . . . . . . . . . . . . 16  |-  (deg ` 
0 p )  =  0
27 0nn0 10238 . . . . . . . . . . . . . . . 16  |-  0  e.  NN0
2826, 27eqeltri 2508 . . . . . . . . . . . . . . 15  |-  (deg ` 
0 p )  e. 
NN0
29 c0ex 9087 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
3029fvconst2 5949 . . . . . . . . . . . . . . 15  |-  ( (deg
`  0 p )  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  (deg `  0 p ) )  =  0 )
3128, 30ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( NN0  X.  { 0 } ) `  (deg `  0 p ) )  =  0
3225, 31eqtri 2458 . . . . . . . . . . . . 13  |-  ( (coeff `  0 p ) `
 (deg `  0 p ) )  =  0
33 ax-1ne0 9061 . . . . . . . . . . . . . 14  |-  1  =/=  0
3433necomi 2688 . . . . . . . . . . . . 13  |-  0  =/=  1
3532, 34eqnetri 2620 . . . . . . . . . . . 12  |-  ( (coeff `  0 p ) `
 (deg `  0 p ) )  =/=  1
36 fveq2 5730 . . . . . . . . . . . . . 14  |-  ( b  =  0 p  -> 
(coeff `  b )  =  (coeff `  0 p
) )
37 fveq2 5730 . . . . . . . . . . . . . 14  |-  ( b  =  0 p  -> 
(deg `  b )  =  (deg `  0 p
) )
3836, 37fveq12d 5736 . . . . . . . . . . . . 13  |-  ( b  =  0 p  -> 
( (coeff `  b
) `  (deg `  b
) )  =  ( (coeff `  0 p
) `  (deg `  0 p ) ) )
3938neeq1d 2616 . . . . . . . . . . . 12  |-  ( b  =  0 p  -> 
( ( (coeff `  b ) `  (deg `  b ) )  =/=  1  <->  ( (coeff ` 
0 p ) `  (deg `  0 p ) )  =/=  1 ) )
4035, 39mpbiri 226 . . . . . . . . . . 11  |-  ( b  =  0 p  -> 
( (coeff `  b
) `  (deg `  b
) )  =/=  1
)
4140necon2i 2653 . . . . . . . . . 10  |-  ( ( (coeff `  b ) `  (deg `  b )
)  =  1  -> 
b  =/=  0 p )
4241ad2antll 711 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  =/=  0 p )
43 eldifsn 3929 . . . . . . . . 9  |-  ( b  e.  ( (Poly `  QQ )  \  { 0 p } )  <->  ( b  e.  (Poly `  QQ )  /\  b  =/=  0 p ) )
4423, 42, 43sylanbrc 647 . . . . . . . 8  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  e.  ( (Poly `  QQ )  \  { 0 p } ) )
45 simprl 734 . . . . . . . 8  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  (
b `  a )  =  0 )
4644, 45jca 520 . . . . . . 7  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  (
b  e.  ( (Poly `  QQ )  \  {
0 p } )  /\  ( b `  a )  =  0 ) )
4746reximi2 2814 . . . . . 6  |-  ( E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 )  ->  E. b  e.  ( (Poly `  QQ )  \  { 0 p } ) ( b `
 a )  =  0 )
4847anim2i 554 . . . . 5  |-  ( ( a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )  -> 
( a  e.  CC  /\ 
E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( b `  a )  =  0 ) )
49 elqaa 20241 . . . . 5  |-  ( a  e.  AA  <->  ( a  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  { 0 p } ) ( b `
 a )  =  0 ) )
5048, 49sylibr 205 . . . 4  |-  ( ( a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )  -> 
a  e.  AA )
5122, 50impbii 182 . . 3  |-  ( a  e.  AA  <->  ( a  e.  CC  /\  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
521, 5, 513bitr4ri 271 . 2  |-  ( a  e.  AA  <->  a  e.  (IntgOver `  QQ ) )
5352eqriv 2435 1  |-  AA  =  (IntgOver `  QQ )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711    \ cdif 3319    C_ wss 3322   {csn 3816    X. cxp 4878   ` cfv 5456   CCcc 8990   0cc0 8992   1c1 8993   NN0cn0 10223   QQcq 10576   0 pc0p 19563  Polycply 20105  coeffccoe 20107  degcdgr 20108   AAcaa 20233  degAAcdgraa 27324  minPolyAAcmpaa 27325  IntgOvercitgo 27341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071
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-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482  df-0p 19564  df-ply 20109  df-coe 20111  df-dgr 20112  df-aa 20234  df-dgraa 27326  df-mpaa 27327  df-itgo 27343
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