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Theorem elaa2 38211
 Description: Elementhood in the set of nonzero algebraic numbers: when is nonzero, the polynomial 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 Polycoeff
Distinct variable group:   ,

Proof of Theorem elaa2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aasscn 23350 . . . 4
2 eldifi 3544 . . . 4
31, 2sseldi 3416 . . 3
4 elaa 23348 . . . . . 6 Poly
52, 4sylib 201 . . . . 5 Poly
65simprd 470 . . . 4 Poly
723ad2ant1 1051 . . . . . 6 Poly
8 eldifsni 4089 . . . . . . 7
983ad2ant1 1051 . . . . . 6 Poly
10 eldifi 3544 . . . . . . 7 Poly Poly
11103ad2ant2 1052 . . . . . 6 Poly Poly
12 eldifsni 4089 . . . . . . 7 Poly
13123ad2ant2 1052 . . . . . 6 Poly
14 simp3 1032 . . . . . 6 Poly
15 fveq2 5879 . . . . . . . . 9 coeff coeff
1615neeq1d 2702 . . . . . . . 8 coeff coeff
1716cbvrabv 3030 . . . . . . 7 coeff coeff
1817infeq1i 8012 . . . . . 6 inf coeff inf coeff
19 oveq1 6315 . . . . . . . 8 inf coeff inf coeff
2019fveq2d 5883 . . . . . . 7 coeff inf coeff coeff inf coeff
2120cbvmptv 4488 . . . . . 6 coeff inf coeff coeff inf coeff
22 eqid 2471 . . . . . 6 deg inf coeff coeff inf coeff deg inf coeff coeff inf coeff
237, 9, 11, 13, 14, 18, 21, 22elaa2lem 38209 . . . . 5 Poly Polycoeff
2423rexlimdv3a 2873 . . . 4 Poly Polycoeff
256, 24mpd 15 . . 3 Polycoeff
263, 25jca 541 . 2 Polycoeff
27 simpl 464 . . . . . . . . 9 Poly coeff Poly
28 fveq2 5879 . . . . . . . . . . . . . . 15 coeff coeff
29 coe0 23289 . . . . . . . . . . . . . . 15 coeff
3028, 29syl6eq 2521 . . . . . . . . . . . . . 14 coeff
3130fveq1d 5881 . . . . . . . . . . . . 13 coeff
32 0nn0 10908 . . . . . . . . . . . . . 14
33 fvconst2g 6134 . . . . . . . . . . . . . 14
3432, 32, 33mp2an 686 . . . . . . . . . . . . 13
3531, 34syl6eq 2521 . . . . . . . . . . . 12 coeff
3635adantl 473 . . . . . . . . . . 11 Poly coeff coeff
37 neneq 2649 . . . . . . . . . . . 12 coeff coeff
3837ad2antlr 741 . . . . . . . . . . 11 Poly coeff coeff
3936, 38pm2.65da 586 . . . . . . . . . 10 Poly coeff
40 elsn 3973 . . . . . . . . . 10
4139, 40sylnibr 312 . . . . . . . . 9 Poly coeff
4227, 41eldifd 3401 . . . . . . . 8 Poly coeff Poly
4342adantrr 731 . . . . . . 7 Poly coeff Poly
44 simprr 774 . . . . . . 7 Poly coeff
4543, 44jca 541 . . . . . 6 Poly coeff Poly
4645reximi2 2851 . . . . 5 Polycoeff Poly
4746anim2i 579 . . . 4 Polycoeff Poly
48 elaa 23348 . . . 4 Poly
4947, 48sylibr 217 . . 3 Polycoeff
50 simpr 468 . . . 4 Polycoeff Polycoeff
51 nfv 1769 . . . . . 6
52 nfre1 2846 . . . . . 6 Polycoeff
5351, 52nfan 2031 . . . . 5 Polycoeff
54 nfv 1769 . . . . 5
55 simpl3r 1086 . . . . . . . . 9 Poly coeff
56 fveq2 5879 . . . . . . . . . . . . . . 15
57 eqid 2471 . . . . . . . . . . . . . . . 16 coeff coeff
5857coefv0 23281 . . . . . . . . . . . . . . 15 Poly coeff
5956, 58sylan9eqr 2527 . . . . . . . . . . . . . 14 Poly coeff
6059adantlr 729 . . . . . . . . . . . . 13 Poly coeff coeff
61 simplr 770 . . . . . . . . . . . . 13 Poly coeff coeff
6260, 61eqnetrd 2710 . . . . . . . . . . . 12 Poly coeff
6362neneqd 2648 . . . . . . . . . . 11 Poly coeff
6463adantlrr 735 . . . . . . . . . 10 Poly coeff
65643adantl1 1186 . . . . . . . . 9 Poly coeff
6655, 65pm2.65da 586 . . . . . . . 8 Poly coeff
67 elsncg 3983 . . . . . . . . . 10
6867biimpa 492 . . . . . . . . 9
69683ad2antl1 1192 . . . . . . . 8 Poly coeff
7066, 69mtand 671 . . . . . . 7 Poly coeff
71703exp 1230 . . . . . 6 Poly coeff
7271adantr 472 . . . . 5 Polycoeff Poly coeff
7353, 54, 72rexlimd 2866 . . . 4 Polycoeff Polycoeff
7450, 73mpd 15 . . 3 Polycoeff
7549, 74eldifd 3401 . 2 Polycoeff
7626, 75impbii 192 1 Polycoeff
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904   wne 2641  wrex 2757  crab 2760   cdif 3387  csn 3959   cmpt 4454   cxp 4837  cfv 5589  (class class class)co 6308  infcinf 7973  cc 9555  cr 9556  cc0 9557   caddc 9560   cmul 9562   clt 9693   cmin 9880  cn0 10893  cz 10961  cfz 11810  cexp 12310  csu 13829  c0p 22706  Polycply 23217  coeffccoe 23219  degcdgr 23220  caa 23346 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224  df-aa 23347 This theorem is referenced by:  etransc  38261
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