Theorem aaitgo 36022

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 2966	. . 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 11272	. . . . 5 |- QQ C_ CC
3		itgoval 36021	. . . . 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 2520	. . 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 23263	. . . . 5 |- ( a e. AA -> a e. CC )
7		mpaacl 36013	. . . . . 6 |- ( a e. AA -> (minPolyAA ` a ) e. (Poly ` QQ ) )
8		mpaaroot 36015	. . . . . 6 |- ( a e. AA -> ( (minPolyAA ` a ) ` a ) = 0 )
9		mpaadgr 36014	. . . . . . . 8 |- ( a e. AA -> (deg ` (minPolyAA ` a ) ) = (degAA ` a ) )
10	9	fveq2d 5867	. . . . . . 7 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (degAA ` a ) ) )
11		mpaamn 36016	. . . . . . 7 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (degAA ` a ) ) = 1 )
12	10, 11	eqtrd 2484	. . . . . 6 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 )
13		fveq1 5862	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( b ` a ) = ( (minPolyAA ` a ) ` a ) )
14	13	eqeq1d 2452	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( b ` a ) = 0 <-> ( (minPolyAA ` a ) ` a ) = 0 ) )
15		fveq2 5863	. . . . . . . . . 10 |- ( b = (minPolyAA ` a ) -> (coeff ` b ) = (coeff ` (minPolyAA ` a ) ) )
16		fveq2 5863	. . . . . . . . . 10 |- ( b = (minPolyAA ` a ) -> (deg ` b ) = (deg ` (minPolyAA ` a ) ) )
17	15, 16	fveq12d 5869	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) )
18	17	eqeq1d 2452	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( (coeff ` b ) ` (deg ` b ) ) = 1 <-> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 ) )
19	14, 18	anbi12d 716	. . . . . . 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 3149	. . . . . 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 1265	. . . . 5 |- ( a e. AA -> E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) )
22	6, 21	jca 535	. . . 4 |- ( a e. AA -> ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) )
23		simpl 459	. . . . . . . . 9 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b e. (Poly ` QQ ) )
24		coe0 23203	. . . . . . . . . . . . . . 15 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
25	24	fveq1i 5864	. . . . . . . . . . . . . 14 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = ( ( NN0 X. { 0 } ) ` (deg ` 0p ) )
26		dgr0 23209	. . . . . . . . . . . . . . . 16 |- (deg ` 0p ) = 0
27		0nn0 10881	. . . . . . . . . . . . . . . 16 |- 0 e. NN0
28	26, 27	eqeltri 2524	. . . . . . . . . . . . . . 15 |- (deg ` 0p ) e. NN0
29		c0ex 9634	. . . . . . . . . . . . . . . 16 |- 0 e. _V
30	29	fvconst2 6118	. . . . . . . . . . . . . . 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 2472	. . . . . . . . . . . . 13 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = 0
33		0ne1 10674	. . . . . . . . . . . . 13 |- 0 =/= 1
34	32, 33	eqnetri 2693	. . . . . . . . . . . 12 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) =/= 1
35		fveq2 5863	. . . . . . . . . . . . . 14 |- ( b = 0p -> (coeff ` b ) = (coeff ` 0p ) )
36		fveq2 5863	. . . . . . . . . . . . . 14 |- ( b = 0p -> (deg ` b ) = (deg ` 0p ) )
37	35, 36	fveq12d 5869	. . . . . . . . . . . . 13 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` 0p ) ` (deg ` 0p ) ) )
38	37	neeq1d 2682	. . . . . . . . . . . 12 |- ( b = 0p -> ( ( (coeff ` b ) ` (deg ` b ) ) =/= 1 <-> ( (coeff ` 0p ) ` (deg ` 0p ) ) =/= 1 ) )
39	34, 38	mpbiri 237	. . . . . . . . . . 11 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) =/= 1 )
40	39	necon2i 2657	. . . . . . . . . 10 |- ( ( (coeff ` b ) ` (deg ` b ) ) = 1 -> b =/= 0p )
41	40	ad2antll 734	. . . . . . . . 9 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b =/= 0p )
42		eldifsn 4096	. . . . . . . . 9 |- ( b e. ( (Poly ` QQ ) \ { 0p } ) <-> ( b e. (Poly ` QQ ) /\ b =/= 0p ) )
43	23, 41, 42	sylanbrc 669	. . . . . . . 8 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b e. ( (Poly ` QQ ) \ { 0p } ) )
44		simprl 763	. . . . . . . 8 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b ` a ) = 0 )
45	43, 44	jca 535	. . . . . . 7 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b e. ( (Poly ` QQ ) \ { 0p } ) /\ ( b ` a ) = 0 ) )
46	45	reximi2 2853	. . . . . 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 572	. . . . 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 23271	. . . . 5 |- ( a e. AA <-> ( a e. CC /\ E. b e. ( (Poly ` QQ ) \ { 0p } ) ( b ` a ) = 0 ) )
49	47, 48	sylibr 216	. . . 4 |- ( ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> a e. AA )
50	22, 49	impbii 191	. . 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 282	. 2 |- ( a e. AA <-> a e. (IntgOver ` QQ ) )
52	51	eqriv 2447	1 |- AA = (IntgOver ` QQ )

Syntax hints:   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   E.wrex 2737   {crab 2740   \ cdif 3400   C_ wss 3403   {csn 3967   X. cxp 4831   ` cfv 5581   CCcc 9534   0cc0 9536   1c1 9537   NN0cn0 10866   QQcq 11261   0pc0p 22620  Polycply 23131  coeffccoe 23133  degcdgr 23134   AAcaa 23260  deg_AAcdgraa 35993  minPolyAAcmpaa 35995  IntgOvercitgo 36017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-rlim 13546  df-sum 13746  df-0p 22621  df-ply 23135  df-coe 23137  df-dgr 23138  df-aa 23261  df-dgraa 35996  df-mpaa 35998  df-itgo 36019

This theorem is referenced by: (None)