Theorem aaitgo 29522

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.

Step	Hyp	Ref	Expression
1		rabid 2900	. . 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 10967	. . . . 5 |- QQ C_ CC
3		itgoval 29521	. . . . 5 |- ( QQ C_ CC -> (IntgOver ` QQ ) = { a e. CC | E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } )
4	2, 3	ax-mp 5	. . . 4 |- (IntgOver ` QQ ) = { a e. CC | E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) }
5	4	eleq2i 2507	. . 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 21786	. . . . 5 |- ( a e. AA -> a e. CC )
7		mpaacl 29513	. . . . . 6 |- ( a e. AA -> (minPolyAA ` a ) e. (Poly ` QQ ) )
8		mpaaroot 29515	. . . . . 6 |- ( a e. AA -> ( (minPolyAA ` a ) ` a ) = 0 )
9		mpaadgr 29514	. . . . . . . 8 |- ( a e. AA -> (deg ` (minPolyAA ` a ) ) = (degAA ` a ) )
10	9	fveq2d 5698	. . . . . . 7 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (degAA ` a ) ) )
11		mpaamn 29516	. . . . . . 7 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (degAA ` a ) ) = 1 )
12	10, 11	eqtrd 2475	. . . . . 6 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 )
13		fveq1 5693	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( b ` a ) = ( (minPolyAA ` a ) ` a ) )
14	13	eqeq1d 2451	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( b ` a ) = 0 <-> ( (minPolyAA ` a ) ` a ) = 0 ) )
15		fveq2 5694	. . . . . . . . . 10 |- ( b = (minPolyAA ` a ) -> (coeff ` b ) = (coeff ` (minPolyAA ` a ) ) )
16		fveq2 5694	. . . . . . . . . 10 |- ( b = (minPolyAA ` a ) -> (deg ` b ) = (deg ` (minPolyAA ` a ) ) )
17	15, 16	fveq12d 5700	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) )
18	17	eqeq1d 2451	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( (coeff ` b ) ` (deg ` b ) ) = 1 <-> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 ) )
19	14, 18	anbi12d 710	. . . . . . 7 |- ( b = (minPolyAA ` a ) -> ( ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) <-> ( ( (minPolyAA ` a ) ` a ) = 0 /\ ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 ) ) )
20	19	rspcev 3076	. . . . . 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 ) )
21	7, 8, 12, 20	syl12anc 1216	. . . . 5 |- ( a e. AA -> E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) )
22	6, 21	jca 532	. . . 4 |- ( a e. AA -> ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) )
23		simpl 457	. . . . . . . . 9 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b e. (Poly ` QQ ) )
24		coe0 21726	. . . . . . . . . . . . . . 15 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
25	24	fveq1i 5695	. . . . . . . . . . . . . 14 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = ( ( NN0 X. { 0 } ) ` (deg ` 0p ) )
26		dgr0 21732	. . . . . . . . . . . . . . . 16 |- (deg ` 0p ) = 0
27		0nn0 10597	. . . . . . . . . . . . . . . 16 |- 0 e. NN0
28	26, 27	eqeltri 2513	. . . . . . . . . . . . . . 15 |- (deg ` 0p ) e. NN0
29		c0ex 9383	. . . . . . . . . . . . . . . 16 |- 0 e. _V
30	29	fvconst2 5936	. . . . . . . . . . . . . . 15 |- ( (deg ` 0p ) e. NN0 -> ( ( NN0 X. { 0 } ) ` (deg ` 0p ) ) = 0 )
31	28, 30	ax-mp 5	. . . . . . . . . . . . . 14 |- ( ( NN0 X. { 0 } ) ` (deg ` 0p ) ) = 0
32	25, 31	eqtri 2463	. . . . . . . . . . . . 13 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = 0
33		0ne1 10392	. . . . . . . . . . . . 13 |- 0 =/= 1
34	32, 33	eqnetri 2628	. . . . . . . . . . . 12 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) =/= 1
35		fveq2 5694	. . . . . . . . . . . . . 14 |- ( b = 0p -> (coeff ` b ) = (coeff ` 0p ) )
36		fveq2 5694	. . . . . . . . . . . . . 14 |- ( b = 0p -> (deg ` b ) = (deg ` 0p ) )
37	35, 36	fveq12d 5700	. . . . . . . . . . . . 13 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` 0p ) ` (deg ` 0p ) ) )
38	37	neeq1d 2624	. . . . . . . . . . . 12 |- ( b = 0p -> ( ( (coeff ` b ) ` (deg ` b ) ) =/= 1 <-> ( (coeff ` 0p ) ` (deg ` 0p ) ) =/= 1 ) )
39	34, 38	mpbiri 233	. . . . . . . . . . 11 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) =/= 1 )
40	39	necon2i 2661	. . . . . . . . . 10 |- ( ( (coeff ` b ) ` (deg ` b ) ) = 1 -> b =/= 0p )
41	40	ad2antll 728	. . . . . . . . 9 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b =/= 0p )
42		eldifsn 4003	. . . . . . . . 9 |- ( b e. ( (Poly ` QQ ) \ { 0p } ) <-> ( b e. (Poly ` QQ ) /\ b =/= 0p ) )
43	23, 41, 42	sylanbrc 664	. . . . . . . 8 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b e. ( (Poly ` QQ ) \ { 0p } ) )
44		simprl 755	. . . . . . . 8 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b ` a ) = 0 )
45	43, 44	jca 532	. . . . . . 7 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b e. ( (Poly ` QQ ) \ { 0p } ) /\ ( b ` a ) = 0 ) )
46	45	reximi2 2825	. . . . . 6 |- ( E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) -> E. b e. ( (Poly ` QQ ) \ { 0p } ) ( b ` a ) = 0 )
47	46	anim2i 569	. . . . 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 ) \ { 0p } ) ( b ` a ) = 0 ) )
48		elqaa 21791	. . . . 5 |- ( a e. AA <-> ( a e. CC /\ E. b e. ( (Poly ` QQ ) \ { 0p } ) ( b ` a ) = 0 ) )
49	47, 48	sylibr 212	. . . 4 |- ( ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> a e. AA )
50	22, 49	impbii 188	. . 3 |- ( a e. AA <-> ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) )
51	1, 5, 50	3bitr4ri 278	. 2 |- ( a e. AA <-> a e. (IntgOver ` QQ ) )
52	51	eqriv 2440	1 |- AA = (IntgOver ` QQ )

Syntax hints:   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2609   E.wrex 2719   {crab 2722   \ cdif 3328   C_ wss 3331   {csn 3880   X. cxp 4841   ` cfv 5421   CCcc 9283   0cc0 9285   1c1 9286   NN0cn0 10582   QQcq 10956   0pc0p 21150  Polycply 21655  coeffccoe 21657  degcdgr 21658   AAcaa 21783  deg_AAcdgraa 29500  minPolyAAcmpaa 29501  IntgOvercitgo 29517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-fz 11441  df-fzo 11552  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-rlim 12970  df-sum 13167  df-0p 21151  df-ply 21659  df-coe 21661  df-dgr 21662  df-aa 21784  df-dgraa 29502  df-mpaa 29503  df-itgo 29519

This theorem is referenced by: (None)