Theorem dgraaub 37718

Description: Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.)

Assertion

Ref	Expression
dgraaub	|- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> (degAA ` A ) <_ (deg ` P ) )

Proof of Theorem dgraaub

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 794	. . . 4 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> A e. CC )
2		eldifsn 4317	. . . . . . 7 |- ( P e. ( (Poly ` QQ ) \ { 0p } ) <-> ( P e. (Poly ` QQ ) /\ P =/= 0p ) )
3	2	biimpri 218	. . . . . 6 |- ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) -> P e. ( (Poly ` QQ ) \ { 0p } ) )
4	3	adantr 481	. . . . 5 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> P e. ( (Poly ` QQ ) \ { 0p } ) )
5		simprr 796	. . . . 5 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> ( P ` A ) = 0 )
6		fveq1 6190	. . . . . . 7 |- ( a = P -> ( a ` A ) = ( P ` A ) )
7	6	eqeq1d 2624	. . . . . 6 |- ( a = P -> ( ( a ` A ) = 0 <-> ( P ` A ) = 0 ) )
8	7	rspcev 3309	. . . . 5 |- ( ( P e. ( (Poly ` QQ ) \ { 0p } ) /\ ( P ` A ) = 0 ) -> E. a e. ( (Poly ` QQ ) \ { 0p } ) ( a ` A ) = 0 )
9	4, 5, 8	syl2anc 693	. . . 4 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> E. a e. ( (Poly ` QQ ) \ { 0p } ) ( a ` A ) = 0 )
10		elqaa 24077	. . . 4 |- ( A e. AA <-> ( A e. CC /\ E. a e. ( (Poly ` QQ ) \ { 0p } ) ( a ` A ) = 0 ) )
11	1, 9, 10	sylanbrc 698	. . 3 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> A e. AA )
12		dgraaval 37714	. . 3 |- ( A e. AA -> (degAA ` A ) = inf ( { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } , RR , < ) )
13	11, 12	syl 17	. 2 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> (degAA ` A ) = inf ( { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } , RR , < ) )
14		ssrab2 3687	. . . 4 |- { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } C_ NN
15		nnuz 11723	. . . 4 |- NN = ( ZZ>= ` 1 )
16	14, 15	sseqtri 3637	. . 3 |- { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } C_ ( ZZ>= ` 1 )
17		dgrnznn 24003	. . . 4 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> (deg ` P ) e. NN )
18		eqid 2622	. . . . . 6 |- (deg ` P ) = (deg ` P )
19	5, 18	jctil 560	. . . . 5 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> ( (deg ` P ) = (deg ` P ) /\ ( P ` A ) = 0 ) )
20		fveq2 6191	. . . . . . . 8 |- ( b = P -> (deg ` b ) = (deg ` P ) )
21	20	eqeq1d 2624	. . . . . . 7 |- ( b = P -> ( (deg ` b ) = (deg ` P ) <-> (deg ` P ) = (deg ` P ) ) )
22		fveq1 6190	. . . . . . . 8 |- ( b = P -> ( b ` A ) = ( P ` A ) )
23	22	eqeq1d 2624	. . . . . . 7 |- ( b = P -> ( ( b ` A ) = 0 <-> ( P ` A ) = 0 ) )
24	21, 23	anbi12d 747	. . . . . 6 |- ( b = P -> ( ( (deg ` b ) = (deg ` P ) /\ ( b ` A ) = 0 ) <-> ( (deg ` P ) = (deg ` P ) /\ ( P ` A ) = 0 ) ) )
25	24	rspcev 3309	. . . . 5 |- ( ( P e. ( (Poly ` QQ ) \ { 0p } ) /\ ( (deg ` P ) = (deg ` P ) /\ ( P ` A ) = 0 ) ) -> E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = (deg ` P ) /\ ( b ` A ) = 0 ) )
26	4, 19, 25	syl2anc 693	. . . 4 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = (deg ` P ) /\ ( b ` A ) = 0 ) )
27		eqeq2 2633	. . . . . . 7 |- ( a = (deg ` P ) -> ( (deg ` b ) = a <-> (deg ` b ) = (deg ` P ) ) )
28	27	anbi1d 741	. . . . . 6 |- ( a = (deg ` P ) -> ( ( (deg ` b ) = a /\ ( b ` A ) = 0 ) <-> ( (deg ` b ) = (deg ` P ) /\ ( b ` A ) = 0 ) ) )
29	28	rexbidv 3052	. . . . 5 |- ( a = (deg ` P ) -> ( E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) <-> E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = (deg ` P ) /\ ( b ` A ) = 0 ) ) )
30	29	elrab 3363	. . . 4 |- ( (deg ` P ) e. { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } <-> ( (deg ` P ) e. NN /\ E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = (deg ` P ) /\ ( b ` A ) = 0 ) ) )
31	17, 26, 30	sylanbrc 698	. . 3 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> (deg ` P ) e. { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } )
32		infssuzle 11771	. . 3 |- ( ( { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } C_ ( ZZ>= ` 1 ) /\ (deg ` P ) e. { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } ) -> inf ( { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } , RR , < ) <_ (deg ` P ) )
33	16, 31, 32	sylancr 695	. 2 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> inf ( { a e. NN | E. b e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` b ) = a /\ ( b ` A ) = 0 ) } , RR , < ) <_ (deg ` P ) )
34	13, 33	eqbrtrd 4675	1 |- ( ( ( P e. (Poly ` QQ ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> (degAA ` A ) <_ (deg ` P ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020   ZZ>=cuz 11687   QQcq 11788   0pc0p 23436  Polycply 23940  degcdgr 23943   AAcaa 24069  deg_AAcdgraa 37710

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

This theorem is referenced by:  dgraa0p  37719