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Theorem qaa 24078
Description: Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
qaa |- ( A e. QQ -> A e. AA )
Proof of Theorem qaa
Dummy variables f x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qcn 11802 . 2 |- ( A e. QQ -> A e. CC )
2 qsscn 11799 . . . . . . 7 |- QQ C_ CC
3 1z 11407 . . . . . . . 8 |- 1 e. ZZ
4 zq 11794 . . . . . . . 8 |- ( 1 e. ZZ -> 1 e. QQ )
53, 4ax-mp 5 . . . . . . 7 |- 1 e. QQ
6 plyid 23965 . . . . . . 7 |- ( ( QQ C_ CC /\ 1 e. QQ ) -> Xp e. (Poly ` QQ ) )
72, 5, 6mp2an 708 . . . . . 6 |- Xp e. (Poly ` QQ )
87a1i 11 . . . . 5 |- ( A e. QQ -> Xp e. (Poly ` QQ ) )
9 plyconst 23962 . . . . . 6 |- ( ( QQ C_ CC /\ A e. QQ ) -> ( CC X. { A } ) e. (Poly ` QQ ) )
102, 9mpan 706 . . . . 5 |- ( A e. QQ -> ( CC X. { A } ) e. (Poly ` QQ ) )
11 qaddcl 11804 . . . . . 6 |- ( ( x e. QQ /\ y e. QQ ) -> ( x + y ) e. QQ )
1211adantl 482 . . . . 5 |- ( ( A e. QQ /\ ( x e. QQ /\ y e. QQ ) ) -> ( x + y ) e. QQ )
13 qmulcl 11806 . . . . . 6 |- ( ( x e. QQ /\ y e. QQ ) -> ( x x. y ) e. QQ )
1413adantl 482 . . . . 5 |- ( ( A e. QQ /\ ( x e. QQ /\ y e. QQ ) ) -> ( x x. y ) e. QQ )
15 qnegcl 11805 . . . . . . 7 |- ( 1 e. QQ -> -u 1 e. QQ )
165, 15ax-mp 5 . . . . . 6 |- -u 1 e. QQ
1716a1i 11 . . . . 5 |- ( A e. QQ -> -u 1 e. QQ )
188, 10, 12, 14, 17plysub 23975 . . . 4 |- ( A e. QQ -> ( Xp oF - ( CC X. { A } ) ) e. (Poly ` QQ ) )
19 peano2cn 10208 . . . . . 6 |- ( A e. CC -> ( A + 1 ) e. CC )
201, 19syl 17 . . . . 5 |- ( A e. QQ -> ( A + 1 ) e. CC )
21 fnresi 6008 . . . . . . . . . . 11 |- ( _I |` CC ) Fn CC
22 df-idp 23945 . . . . . . . . . . . 12 |- Xp = ( _I |` CC )
2322fneq1i 5985 . . . . . . . . . . 11 |- ( Xp Fn CC <-> ( _I |` CC ) Fn CC )
2421, 23mpbir 221 . . . . . . . . . 10 |- Xp Fn CC
2524a1i 11 . . . . . . . . 9 |- ( A e. QQ -> Xp Fn CC )
26 fnconstg 6093 . . . . . . . . 9 |- ( A e. QQ -> ( CC X. { A } ) Fn CC )
27 cnex 10017 . . . . . . . . . 10 |- CC e. _V
2827a1i 11 . . . . . . . . 9 |- ( A e. QQ -> CC e. _V )
29 inidm 3822 . . . . . . . . 9 |- ( CC i^i CC ) = CC
3022fveq1i 6192 . . . . . . . . . . 11 |- ( Xp ` ( A + 1 ) ) = ( ( _I |` CC ) ` ( A + 1 ) )
31 fvresi 6439 . . . . . . . . . . 11 |- ( ( A + 1 ) e. CC -> ( ( _I |` CC ) ` ( A + 1 ) ) = ( A + 1 ) )
3230, 31syl5eq 2668 . . . . . . . . . 10 |- ( ( A + 1 ) e. CC -> ( Xp ` ( A + 1 ) ) = ( A + 1 ) )
3332adantl 482 . . . . . . . . 9 |- ( ( A e. QQ /\ ( A + 1 ) e. CC ) -> ( Xp ` ( A + 1 ) ) = ( A + 1 ) )
34 fvconst2g 6467 . . . . . . . . 9 |- ( ( A e. QQ /\ ( A + 1 ) e. CC ) -> ( ( CC X. { A } ) ` ( A + 1 ) ) = A )
3525, 26, 28, 28, 29, 33, 34ofval 6906 . . . . . . . 8 |- ( ( A e. QQ /\ ( A + 1 ) e. CC ) -> ( ( Xp oF - ( CC X. { A } ) ) ` ( A + 1 ) ) = ( ( A + 1 ) - A ) )
3620, 35mpdan 702 . . . . . . 7 |- ( A e. QQ -> ( ( Xp oF - ( CC X. { A } ) ) ` ( A + 1 ) ) = ( ( A + 1 ) - A ) )
37 ax-1cn 9994 . . . . . . . 8 |- 1 e. CC
38 pncan2 10288 . . . . . . . 8 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - A ) = 1 )
391, 37, 38sylancl 694 . . . . . . 7 |- ( A e. QQ -> ( ( A + 1 ) - A ) = 1 )
4036, 39eqtrd 2656 . . . . . 6 |- ( A e. QQ -> ( ( Xp oF - ( CC X. { A } ) ) ` ( A + 1 ) ) = 1 )
41 ax-1ne0 10005 . . . . . . 7 |- 1 =/= 0
4241a1i 11 . . . . . 6 |- ( A e. QQ -> 1 =/= 0 )
4340, 42eqnetrd 2861 . . . . 5 |- ( A e. QQ -> ( ( Xp oF - ( CC X. { A } ) ) ` ( A + 1 ) ) =/= 0 )
44 ne0p 23963 . . . . 5 |- ( ( ( A + 1 ) e. CC /\ ( ( Xp oF - ( CC X. { A } ) ) ` ( A + 1 ) ) =/= 0 ) -> ( Xp oF - ( CC X. { A } ) ) =/= 0p )
4520, 43, 44syl2anc 693 . . . 4 |- ( A e. QQ -> ( Xp oF - ( CC X. { A } ) ) =/= 0p )
46 eldifsn 4317 . . . 4 |- ( ( Xp oF - ( CC X. { A } ) ) e. ( (Poly ` QQ ) \ { 0p } ) <-> ( ( Xp oF - ( CC X. { A } ) ) e. (Poly ` QQ ) /\ ( Xp oF - ( CC X. { A } ) ) =/= 0p ) )
4718, 45, 46sylanbrc 698 . . 3 |- ( A e. QQ -> ( Xp oF - ( CC X. { A } ) ) e. ( (Poly ` QQ ) \ { 0p } ) )
4822fveq1i 6192 . . . . . . . 8 |- ( Xp ` A ) = ( ( _I |` CC ) ` A )
49 fvresi 6439 . . . . . . . 8 |- ( A e. CC -> ( ( _I |` CC ) ` A ) = A )
5048, 49syl5eq 2668 . . . . . . 7 |- ( A e. CC -> ( Xp ` A ) = A )
5150adantl 482 . . . . . 6 |- ( ( A e. QQ /\ A e. CC ) -> ( Xp ` A ) = A )
52 fvconst2g 6467 . . . . . 6 |- ( ( A e. QQ /\ A e. CC ) -> ( ( CC X. { A } ) ` A ) = A )
5325, 26, 28, 28, 29, 51, 52ofval 6906 . . . . 5 |- ( ( A e. QQ /\ A e. CC ) -> ( ( Xp oF - ( CC X. { A } ) ) ` A ) = ( A - A ) )
541, 53mpdan 702 . . . 4 |- ( A e. QQ -> ( ( Xp oF - ( CC X. { A } ) ) ` A ) = ( A - A ) )
551subidd 10380 . . . 4 |- ( A e. QQ -> ( A - A ) = 0 )
5654, 55eqtrd 2656 . . 3 |- ( A e. QQ -> ( ( Xp oF - ( CC X. { A } ) ) ` A ) = 0 )
57 fveq1 6190 . . . . 5 |- ( f = ( Xp oF - ( CC X. { A } ) ) -> ( f ` A ) = ( ( Xp oF - ( CC X. { A } ) ) ` A ) )
5857eqeq1d 2624 . . . 4 |- ( f = ( Xp oF - ( CC X. { A } ) ) -> ( ( f ` A ) = 0 <-> ( ( Xp oF - ( CC X. { A } ) ) ` A ) = 0 ) )
5958rspcev 3309 . . 3 |- ( ( ( Xp oF - ( CC X. { A } ) ) e. ( (Poly ` QQ ) \ { 0p } ) /\ ( ( Xp oF - ( CC X. { A } ) ) ` A ) = 0 ) -> E. f e. ( (Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 )
6047, 56, 59syl2anc 693 . 2 |- ( A e. QQ -> E. f e. ( (Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 )
61 elqaa 24077 . 2 |- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 ) )
621, 60, 61sylanbrc 698 1 |- ( A e. QQ -> A e. AA )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   _I cid 5023   X. cxp 5112   |` cres 5116   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   ZZcz 11377   QQcq 11788   0pc0p 23436  Polycply 23940   Xpcidp 23941   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-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-idp 23945  df-coe 23946  df-dgr 23947  df-aa 24070
This theorem is referenced by:  qssaa  24079
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