Theorem elaa2 40451

Description: Elementhood in the set of nonzero algebraic numbers: when A is nonzero, the polynomial f can be chosen with a nonzero constant term. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Proof shortened by AV, 1-Oct-2020.)

Assertion

Ref	Expression
elaa2	|- ( A e. ( AA \ { 0 } ) <-> ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )

Distinct variable group:   A, f

Proof of Theorem elaa2

Dummy variables k g z j m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aasscn 24073	. . . 4 |- AA C_ CC
2		eldifi 3732	. . . 4 |- ( A e. ( AA \ { 0 } ) -> A e. AA )
3	1, 2	sseldi 3601	. . 3 |- ( A e. ( AA \ { 0 } ) -> A e. CC )
4		elaa 24071	. . . . . 6 |- ( A e. AA <-> ( A e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) )
5	2, 4	sylib 208	. . . . 5 |- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) )
6	5	simprd 479	. . . 4 |- ( A e. ( AA \ { 0 } ) -> E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 )
7	2	3ad2ant1 1082	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A e. AA )
8		eldifsni 4320	. . . . . . 7 |- ( A e. ( AA \ { 0 } ) -> A =/= 0 )
9	8	3ad2ant1 1082	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A =/= 0 )
10		eldifi 3732	. . . . . . 7 |- ( g e. ( (Poly ` ZZ ) \ { 0p } ) -> g e. (Poly ` ZZ ) )
11	10	3ad2ant2 1083	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g e. (Poly ` ZZ ) )
12		eldifsni 4320	. . . . . . 7 |- ( g e. ( (Poly ` ZZ ) \ { 0p } ) -> g =/= 0p )
13	12	3ad2ant2 1083	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g =/= 0p )
14		simp3 1063	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> ( g ` A ) = 0 )
15		fveq2 6191	. . . . . . . . 9 |- ( m = n -> ( (coeff ` g ) ` m ) = ( (coeff ` g ) ` n ) )
16	15	neeq1d 2853	. . . . . . . 8 |- ( m = n -> ( ( (coeff ` g ) ` m ) =/= 0 <-> ( (coeff ` g ) ` n ) =/= 0 ) )
17	16	cbvrabv 3199	. . . . . . 7 |- { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } = { n e. NN0 | ( (coeff ` g ) ` n ) =/= 0 }
18	17	infeq1i 8384	. . . . . 6 |- inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) = inf ( { n e. NN0 | ( (coeff ` g ) ` n ) =/= 0 } , RR , < )
19		oveq1 6657	. . . . . . . 8 |- ( j = k -> ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) = ( k + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) )
20	19	fveq2d 6195	. . . . . . 7 |- ( j = k -> ( (coeff ` g ) ` ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) = ( (coeff ` g ) ` ( k + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) )
21	20	cbvmptv 4750	. . . . . 6 |- ( j e. NN0 |-> ( (coeff ` g ) ` ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) = ( k e. NN0 |-> ( (coeff ` g ) ` ( k + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) )
22		eqid 2622	. . . . . 6 |- ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` g ) - inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ( ( ( j e. NN0 |-> ( (coeff ` g ) ` ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` g ) - inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ( ( ( j e. NN0 |-> ( (coeff ` g ) ` ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) ` k ) x. ( z ^ k ) ) )
23	7, 9, 11, 13, 14, 18, 21, 22	elaa2lem 40450	. . . . 5 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
24	23	rexlimdv3a 3033	. . . 4 |- ( A e. ( AA \ { 0 } ) -> ( E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )
25	6, 24	mpd 15	. . 3 |- ( A e. ( AA \ { 0 } ) -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
26	3, 25	jca 554	. 2 |- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )
27		simpl 473	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> f e. (Poly ` ZZ ) )
28		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( f = 0p -> (coeff ` f ) = (coeff ` 0p ) )
29		coe0 24012	. . . . . . . . . . . . . . 15 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
30	28, 29	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( f = 0p -> (coeff ` f ) = ( NN0 X. { 0 } ) )
31	30	fveq1d 6193	. . . . . . . . . . . . 13 |- ( f = 0p -> ( (coeff ` f ) ` 0 ) = ( ( NN0 X. { 0 } ) ` 0 ) )
32		0nn0 11307	. . . . . . . . . . . . . 14 |- 0 e. NN0
33		fvconst2g 6467	. . . . . . . . . . . . . 14 |- ( ( 0 e. NN0 /\ 0 e. NN0 ) -> ( ( NN0 X. { 0 } ) ` 0 ) = 0 )
34	32, 32, 33	mp2an 708	. . . . . . . . . . . . 13 |- ( ( NN0 X. { 0 } ) ` 0 ) = 0
35	31, 34	syl6eq 2672	. . . . . . . . . . . 12 |- ( f = 0p -> ( (coeff ` f ) ` 0 ) = 0 )
36	35	adantl 482	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> ( (coeff ` f ) ` 0 ) = 0 )
37		neneq 2800	. . . . . . . . . . . 12 |- ( ( (coeff ` f ) ` 0 ) =/= 0 -> -. ( (coeff ` f ) ` 0 ) = 0 )
38	37	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> -. ( (coeff ` f ) ` 0 ) = 0 )
39	36, 38	pm2.65da 600	. . . . . . . . . 10 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> -. f = 0p )
40		velsn 4193	. . . . . . . . . 10 |- ( f e. { 0p } <-> f = 0p )
41	39, 40	sylnibr 319	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> -. f e. { 0p } )
42	27, 41	eldifd 3585	. . . . . . . 8 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
43	42	adantrr 753	. . . . . . 7 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
44		simprr 796	. . . . . . 7 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f ` A ) = 0 )
45	43, 44	jca 554	. . . . . 6 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) )
46	45	reximi2 3010	. . . . 5 |- ( E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 )
47	46	anim2i 593	. . . 4 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
48		elaa 24071	. . . 4 |- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
49	47, 48	sylibr 224	. . 3 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. AA )
50		simpr 477	. . . 4 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
51		nfv 1843	. . . . . 6 |- F/ f A e. CC
52		nfre1 3005	. . . . . 6 |- F/ f E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 )
53	51, 52	nfan 1828	. . . . 5 |- F/ f ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
54		nfv 1843	. . . . 5 |- F/ f -. A e. { 0 }
55		simpl3r 1117	. . . . . . . . 9 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> ( f ` A ) = 0 )
56		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( A = 0 -> ( f ` A ) = ( f ` 0 ) )
57		eqid 2622	. . . . . . . . . . . . . . . 16 |- (coeff ` f ) = (coeff ` f )
58	57	coefv0 24004	. . . . . . . . . . . . . . 15 |- ( f e. (Poly ` ZZ ) -> ( f ` 0 ) = ( (coeff ` f ) ` 0 ) )
59	56, 58	sylan9eqr 2678	. . . . . . . . . . . . . 14 |- ( ( f e. (Poly ` ZZ ) /\ A = 0 ) -> ( f ` A ) = ( (coeff ` f ) ` 0 ) )
60	59	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) = ( (coeff ` f ) ` 0 ) )
61		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( (coeff ` f ) ` 0 ) =/= 0 )
62	60, 61	eqnetrd 2861	. . . . . . . . . . . 12 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) =/= 0 )
63	62	neneqd 2799	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
64	63	adantlrr 757	. . . . . . . . . 10 |- ( ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
65	64	3adantl1 1217	. . . . . . . . 9 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
66	55, 65	pm2.65da 600	. . . . . . . 8 |- ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A = 0 )
67		elsng 4191	. . . . . . . . . 10 |- ( A e. CC -> ( A e. { 0 } <-> A = 0 ) )
68	67	biimpa 501	. . . . . . . . 9 |- ( ( A e. CC /\ A e. { 0 } ) -> A = 0 )
69	68	3ad2antl1 1223	. . . . . . . 8 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A e. { 0 } ) -> A = 0 )
70	66, 69	mtand 691	. . . . . . 7 |- ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A e. { 0 } )
71	70	3exp 1264	. . . . . 6 |- ( A e. CC -> ( f e. (Poly ` ZZ ) -> ( ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) )
72	71	adantr 481	. . . . 5 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f e. (Poly ` ZZ ) -> ( ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) )
73	53, 54, 72	rexlimd 3026	. . . 4 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) )
74	50, 73	mpd 15	. . 3 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A e. { 0 } )
75	49, 74	eldifd 3585	. 2 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. ( AA \ { 0 } ) )
76	26, 75	impbii 199	1 |- ( A e. ( AA \ { 0 } ) <-> ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   {csn 4177   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   NN0cn0 11292   ZZcz 11377   ...cfz 12326   ^cexp 12860   sum_csu 14416   0pc0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943   AAcaa 24069

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-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  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

This theorem is referenced by:  etransc  40500