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Theorem plyremlem 24059
Description: Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
plyrem.1 |- G = ( Xp oF - ( CC X. { A } ) )
Assertion
Ref Expression
plyremlem |- ( A e. CC -> ( G e. (Poly ` CC ) /\ (deg ` G ) = 1 /\ ( `' G " { 0 } ) = { A } ) )
Proof of Theorem plyremlem
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 plyrem.1 . . 3 |- G = ( Xp oF - ( CC X. { A } ) )
2 ssid 3624 . . . . 5 |- CC C_ CC
3 ax-1cn 9994 . . . . 5 |- 1 e. CC
4 plyid 23965 . . . . 5 |- ( ( CC C_ CC /\ 1 e. CC ) -> Xp e. (Poly ` CC ) )
52, 3, 4mp2an 708 . . . 4 |- Xp e. (Poly ` CC )
6 plyconst 23962 . . . . 5 |- ( ( CC C_ CC /\ A e. CC ) -> ( CC X. { A } ) e. (Poly ` CC ) )
72, 6mpan 706 . . . 4 |- ( A e. CC -> ( CC X. { A } ) e. (Poly ` CC ) )
8 plysubcl 23978 . . . 4 |- ( ( Xp e. (Poly ` CC ) /\ ( CC X. { A } ) e. (Poly ` CC ) ) -> ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) )
95, 7, 8sylancr 695 . . 3 |- ( A e. CC -> ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) )
101, 9syl5eqel 2705 . 2 |- ( A e. CC -> G e. (Poly ` CC ) )
11 negcl 10281 . . . . . . . . 9 |- ( A e. CC -> -u A e. CC )
12 addcom 10222 . . . . . . . . 9 |- ( ( -u A e. CC /\ z e. CC ) -> ( -u A + z ) = ( z + -u A ) )
1311, 12sylan 488 . . . . . . . 8 |- ( ( A e. CC /\ z e. CC ) -> ( -u A + z ) = ( z + -u A ) )
14 negsub 10329 . . . . . . . . 9 |- ( ( z e. CC /\ A e. CC ) -> ( z + -u A ) = ( z - A ) )
1514ancoms 469 . . . . . . . 8 |- ( ( A e. CC /\ z e. CC ) -> ( z + -u A ) = ( z - A ) )
1613, 15eqtrd 2656 . . . . . . 7 |- ( ( A e. CC /\ z e. CC ) -> ( -u A + z ) = ( z - A ) )
1716mpteq2dva 4744 . . . . . 6 |- ( A e. CC -> ( z e. CC |-> ( -u A + z ) ) = ( z e. CC |-> ( z - A ) ) )
18 cnex 10017 . . . . . . . 8 |- CC e. _V
1918a1i 11 . . . . . . 7 |- ( A e. CC -> CC e. _V )
20 negex 10279 . . . . . . . 8 |- -u A e. _V
2120a1i 11 . . . . . . 7 |- ( ( A e. CC /\ z e. CC ) -> -u A e. _V )
22 simpr 477 . . . . . . 7 |- ( ( A e. CC /\ z e. CC ) -> z e. CC )
23 fconstmpt 5163 . . . . . . . 8 |- ( CC X. { -u A } ) = ( z e. CC |-> -u A )
2423a1i 11 . . . . . . 7 |- ( A e. CC -> ( CC X. { -u A } ) = ( z e. CC |-> -u A ) )
25 df-idp 23945 . . . . . . . . 9 |- Xp = ( _I |` CC )
26 mptresid 5456 . . . . . . . . 9 |- ( z e. CC |-> z ) = ( _I |` CC )
2725, 26eqtr4i 2647 . . . . . . . 8 |- Xp = ( z e. CC |-> z )
2827a1i 11 . . . . . . 7 |- ( A e. CC -> Xp = ( z e. CC |-> z ) )
2919, 21, 22, 24, 28offval2 6914 . . . . . 6 |- ( A e. CC -> ( ( CC X. { -u A } ) oF + Xp ) = ( z e. CC |-> ( -u A + z ) ) )
30 simpl 473 . . . . . . 7 |- ( ( A e. CC /\ z e. CC ) -> A e. CC )
31 fconstmpt 5163 . . . . . . . 8 |- ( CC X. { A } ) = ( z e. CC |-> A )
3231a1i 11 . . . . . . 7 |- ( A e. CC -> ( CC X. { A } ) = ( z e. CC |-> A ) )
3319, 22, 30, 28, 32offval2 6914 . . . . . 6 |- ( A e. CC -> ( Xp oF - ( CC X. { A } ) ) = ( z e. CC |-> ( z - A ) ) )
3417, 29, 333eqtr4d 2666 . . . . 5 |- ( A e. CC -> ( ( CC X. { -u A } ) oF + Xp ) = ( Xp oF - ( CC X. { A } ) ) )
3534, 1syl6eqr 2674 . . . 4 |- ( A e. CC -> ( ( CC X. { -u A } ) oF + Xp ) = G )
3635fveq2d 6195 . . 3 |- ( A e. CC -> (deg ` ( ( CC X. { -u A } ) oF + Xp ) ) = (deg ` G ) )
37 plyconst 23962 . . . . 5 |- ( ( CC C_ CC /\ -u A e. CC ) -> ( CC X. { -u A } ) e. (Poly ` CC ) )
382, 11, 37sylancr 695 . . . 4 |- ( A e. CC -> ( CC X. { -u A } ) e. (Poly ` CC ) )
395a1i 11 . . . 4 |- ( A e. CC -> Xp e. (Poly ` CC ) )
40 0dgr 24001 . . . . . 6 |- ( -u A e. CC -> (deg ` ( CC X. { -u A } ) ) = 0 )
4111, 40syl 17 . . . . 5 |- ( A e. CC -> (deg ` ( CC X. { -u A } ) ) = 0 )
42 0lt1 10550 . . . . 5 |- 0 < 1
4341, 42syl6eqbr 4692 . . . 4 |- ( A e. CC -> (deg ` ( CC X. { -u A } ) ) < 1 )
44 eqid 2622 . . . . 5 |- (deg ` ( CC X. { -u A } ) ) = (deg ` ( CC X. { -u A } ) )
45 dgrid 24020 . . . . . 6 |- (deg ` Xp ) = 1
4645eqcomi 2631 . . . . 5 |- 1 = (deg ` Xp )
4744, 46dgradd2 24024 . . . 4 |- ( ( ( CC X. { -u A } ) e. (Poly ` CC ) /\ Xp e. (Poly ` CC ) /\ (deg ` ( CC X. { -u A } ) ) < 1 ) -> (deg ` ( ( CC X. { -u A } ) oF + Xp ) ) = 1 )
4838, 39, 43, 47syl3anc 1326 . . 3 |- ( A e. CC -> (deg ` ( ( CC X. { -u A } ) oF + Xp ) ) = 1 )
4936, 48eqtr3d 2658 . 2 |- ( A e. CC -> (deg ` G ) = 1 )
501, 33syl5eq 2668 . . . . . . . . . . 11 |- ( A e. CC -> G = ( z e. CC |-> ( z - A ) ) )
5150fveq1d 6193 . . . . . . . . . 10 |- ( A e. CC -> ( G ` z ) = ( ( z e. CC |-> ( z - A ) ) ` z ) )
5251adantr 481 . . . . . . . . 9 |- ( ( A e. CC /\ z e. CC ) -> ( G ` z ) = ( ( z e. CC |-> ( z - A ) ) ` z ) )
53 ovex 6678 . . . . . . . . . 10 |- ( z - A ) e. _V
54 eqid 2622 . . . . . . . . . . 11 |- ( z e. CC |-> ( z - A ) ) = ( z e. CC |-> ( z - A ) )
5554fvmpt2 6291 . . . . . . . . . 10 |- ( ( z e. CC /\ ( z - A ) e. _V ) -> ( ( z e. CC |-> ( z - A ) ) ` z ) = ( z - A ) )
5622, 53, 55sylancl 694 . . . . . . . . 9 |- ( ( A e. CC /\ z e. CC ) -> ( ( z e. CC |-> ( z - A ) ) ` z ) = ( z - A ) )
5752, 56eqtrd 2656 . . . . . . . 8 |- ( ( A e. CC /\ z e. CC ) -> ( G ` z ) = ( z - A ) )
5857eqeq1d 2624 . . . . . . 7 |- ( ( A e. CC /\ z e. CC ) -> ( ( G ` z ) = 0 <-> ( z - A ) = 0 ) )
59 subeq0 10307 . . . . . . . 8 |- ( ( z e. CC /\ A e. CC ) -> ( ( z - A ) = 0 <-> z = A ) )
6059ancoms 469 . . . . . . 7 |- ( ( A e. CC /\ z e. CC ) -> ( ( z - A ) = 0 <-> z = A ) )
6158, 60bitrd 268 . . . . . 6 |- ( ( A e. CC /\ z e. CC ) -> ( ( G ` z ) = 0 <-> z = A ) )
6261pm5.32da 673 . . . . 5 |- ( A e. CC -> ( ( z e. CC /\ ( G ` z ) = 0 ) <-> ( z e. CC /\ z = A ) ) )
63 plyf 23954 . . . . . 6 |- ( G e. (Poly ` CC ) -> G : CC --> CC )
64 ffn 6045 . . . . . 6 |- ( G : CC --> CC -> G Fn CC )
65 fniniseg 6338 . . . . . 6 |- ( G Fn CC -> ( z e. ( `' G " { 0 } ) <-> ( z e. CC /\ ( G ` z ) = 0 ) ) )
6610, 63, 64, 654syl 19 . . . . 5 |- ( A e. CC -> ( z e. ( `' G " { 0 } ) <-> ( z e. CC /\ ( G ` z ) = 0 ) ) )
67 eleq1a 2696 . . . . . 6 |- ( A e. CC -> ( z = A -> z e. CC ) )
6867pm4.71rd 667 . . . . 5 |- ( A e. CC -> ( z = A <-> ( z e. CC /\ z = A ) ) )
6962, 66, 683bitr4d 300 . . . 4 |- ( A e. CC -> ( z e. ( `' G " { 0 } ) <-> z = A ) )
70 velsn 4193 . . . 4 |- ( z e. { A } <-> z = A )
7169, 70syl6bbr 278 . . 3 |- ( A e. CC -> ( z e. ( `' G " { 0 } ) <-> z e. { A } ) )
7271eqrdv 2620 . 2 |- ( A e. CC -> ( `' G " { 0 } ) = { A } )
7310, 49, 723jca 1242 1 |- ( A e. CC -> ( G e. (Poly ` CC ) /\ (deg ` G ) = 1 /\ ( `' G " { 0 } ) = { A } ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   -ucneg 10267  Polycply 23940   Xpcidp 23941  degcdgr 23943
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-idp 23945  df-coe 23946  df-dgr 23947
This theorem is referenced by:  plyrem  24060  facth  24061  fta1lem  24062  vieta1lem1  24065  vieta1lem2  24066  taylply2  24122  ftalem7  24805
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