Theorem aaitgo 37732

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 3116	. . 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 11799	. . . . 5 |- QQ C_ CC
3		itgoval 37731	. . . . 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 2693	. . 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 24072	. . . . 5 |- ( a e. AA -> a e. CC )
7		mpaacl 37723	. . . . . 6 |- ( a e. AA -> (minPolyAA ` a ) e. (Poly ` QQ ) )
8		mpaaroot 37725	. . . . . 6 |- ( a e. AA -> ( (minPolyAA ` a ) ` a ) = 0 )
9		mpaadgr 37724	. . . . . . . 8 |- ( a e. AA -> (deg ` (minPolyAA ` a ) ) = (degAA ` a ) )
10	9	fveq2d 6195	. . . . . . 7 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (degAA ` a ) ) )
11		mpaamn 37726	. . . . . . 7 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (degAA ` a ) ) = 1 )
12	10, 11	eqtrd 2656	. . . . . 6 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 )
13		fveq1 6190	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( b ` a ) = ( (minPolyAA ` a ) ` a ) )
14	13	eqeq1d 2624	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( b ` a ) = 0 <-> ( (minPolyAA ` a ) ` a ) = 0 ) )
15		fveq2 6191	. . . . . . . . . 10 |- ( b = (minPolyAA ` a ) -> (coeff ` b ) = (coeff ` (minPolyAA ` a ) ) )
16		fveq2 6191	. . . . . . . . . 10 |- ( b = (minPolyAA ` a ) -> (deg ` b ) = (deg ` (minPolyAA ` a ) ) )
17	15, 16	fveq12d 6197	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) )
18	17	eqeq1d 2624	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( (coeff ` b ) ` (deg ` b ) ) = 1 <-> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 ) )
19	14, 18	anbi12d 747	. . . . . . 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 3309	. . . . . 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 1324	. . . . 5 |- ( a e. AA -> E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) )
22	6, 21	jca 554	. . . 4 |- ( a e. AA -> ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) )
23		simpl 473	. . . . . . . . 9 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b e. (Poly ` QQ ) )
24		coe0 24012	. . . . . . . . . . . . . . 15 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
25	24	fveq1i 6192	. . . . . . . . . . . . . 14 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = ( ( NN0 X. { 0 } ) ` (deg ` 0p ) )
26		dgr0 24018	. . . . . . . . . . . . . . . 16 |- (deg ` 0p ) = 0
27		0nn0 11307	. . . . . . . . . . . . . . . 16 |- 0 e. NN0
28	26, 27	eqeltri 2697	. . . . . . . . . . . . . . 15 |- (deg ` 0p ) e. NN0
29		c0ex 10034	. . . . . . . . . . . . . . . 16 |- 0 e. _V
30	29	fvconst2 6469	. . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . 13 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = 0
33		0ne1 11088	. . . . . . . . . . . . 13 |- 0 =/= 1
34	32, 33	eqnetri 2864	. . . . . . . . . . . 12 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) =/= 1
35		fveq2 6191	. . . . . . . . . . . . . 14 |- ( b = 0p -> (coeff ` b ) = (coeff ` 0p ) )
36		fveq2 6191	. . . . . . . . . . . . . 14 |- ( b = 0p -> (deg ` b ) = (deg ` 0p ) )
37	35, 36	fveq12d 6197	. . . . . . . . . . . . 13 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` 0p ) ` (deg ` 0p ) ) )
38	37	neeq1d 2853	. . . . . . . . . . . 12 |- ( b = 0p -> ( ( (coeff ` b ) ` (deg ` b ) ) =/= 1 <-> ( (coeff ` 0p ) ` (deg ` 0p ) ) =/= 1 ) )
39	34, 38	mpbiri 248	. . . . . . . . . . 11 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) =/= 1 )
40	39	necon2i 2828	. . . . . . . . . 10 |- ( ( (coeff ` b ) ` (deg ` b ) ) = 1 -> b =/= 0p )
41	40	ad2antll 765	. . . . . . . . 9 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b =/= 0p )
42		eldifsn 4317	. . . . . . . . 9 |- ( b e. ( (Poly ` QQ ) \ { 0p } ) <-> ( b e. (Poly ` QQ ) /\ b =/= 0p ) )
43	23, 41, 42	sylanbrc 698	. . . . . . . 8 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b e. ( (Poly ` QQ ) \ { 0p } ) )
44		simprl 794	. . . . . . . 8 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b ` a ) = 0 )
45	43, 44	jca 554	. . . . . . 7 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b e. ( (Poly ` QQ ) \ { 0p } ) /\ ( b ` a ) = 0 ) )
46	45	reximi2 3010	. . . . . 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 593	. . . . 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 24077	. . . . 5 |- ( a e. AA <-> ( a e. CC /\ E. b e. ( (Poly ` QQ ) \ { 0p } ) ( b ` a ) = 0 ) )
49	47, 48	sylibr 224	. . . 4 |- ( ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> a e. AA )
50	22, 49	impbii 199	. . 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 293	. 2 |- ( a e. AA <-> a e. (IntgOver ` QQ ) )
52	51	eqriv 2619	1 |- AA = (IntgOver ` QQ )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   {csn 4177   X. cxp 5112   ` cfv 5888   CCcc 9934   0cc0 9936   1c1 9937   NN0cn0 11292   QQcq 11788   0pc0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943   AAcaa 24069  deg_AAcdgraa 37710  minPolyAAcmpaa 37711  IntgOvercitgo 37727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947  df-aa 24070  df-dgraa 37712  df-mpaa 37713  df-itgo 37729

This theorem is referenced by: (None)