Theorem uc1pmon1p 23911

Description: Make a unitic polynomial monic by multiplying a factor to normalize the leading coefficient. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Hypotheses

Ref	Expression
uc1pmon1p.c	|- C = (Unic1p ` R )
uc1pmon1p.m	|- M = (Monic1p ` R )
uc1pmon1p.p	|- P = (Poly1 ` R )
uc1pmon1p.t	|- .x. = ( .r ` P )
uc1pmon1p.a	|- A = (algSc ` P )
uc1pmon1p.d	|- D = ( deg1 ` R )
uc1pmon1p.i	|- I = ( invr ` R )

Assertion

Ref	Expression
uc1pmon1p	|- ( ( R e. Ring /\ X e. C ) -> ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) e. M )

Proof of Theorem uc1pmon1p

Step	Hyp	Ref	Expression
1		uc1pmon1p.p	. . . . 5 |- P = (Poly1 ` R )
2	1	ply1ring 19618	. . . 4 |- ( R e. Ring -> P e. Ring )
3	2	adantr 481	. . 3 |- ( ( R e. Ring /\ X e. C ) -> P e. Ring )
4		uc1pmon1p.a	. . . . . 6 |- A = (algSc ` P )
5		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
6		eqid 2622	. . . . . 6 |- ( Base ` P ) = ( Base ` P )
7	1, 4, 5, 6	ply1sclf 19655	. . . . 5 |- ( R e. Ring -> A : ( Base ` R ) --> ( Base ` P ) )
8	7	adantr 481	. . . 4 |- ( ( R e. Ring /\ X e. C ) -> A : ( Base ` R ) --> ( Base ` P ) )
9		uc1pmon1p.d	. . . . . 6 |- D = ( deg1 ` R )
10		eqid 2622	. . . . . 6 |- (Unit ` R ) = (Unit ` R )
11		uc1pmon1p.c	. . . . . 6 |- C = (Unic1p ` R )
12	9, 10, 11	uc1pldg 23908	. . . . 5 |- ( X e. C -> ( (coe1 ` X ) ` ( D ` X ) ) e. (Unit ` R ) )
13		uc1pmon1p.i	. . . . . 6 |- I = ( invr ` R )
14	10, 13, 5	ringinvcl 18676	. . . . 5 |- ( ( R e. Ring /\ ( (coe1 ` X ) ` ( D ` X ) ) e. (Unit ` R ) ) -> ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) e. ( Base ` R ) )
15	12, 14	sylan2 491	. . . 4 |- ( ( R e. Ring /\ X e. C ) -> ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) e. ( Base ` R ) )
16	8, 15	ffvelrnd 6360	. . 3 |- ( ( R e. Ring /\ X e. C ) -> ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) e. ( Base ` P ) )
17	1, 6, 11	uc1pcl 23903	. . . 4 |- ( X e. C -> X e. ( Base ` P ) )
18	17	adantl 482	. . 3 |- ( ( R e. Ring /\ X e. C ) -> X e. ( Base ` P ) )
19		uc1pmon1p.t	. . . 4 |- .x. = ( .r ` P )
20	6, 19	ringcl 18561	. . 3 |- ( ( P e. Ring /\ ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) e. ( Base ` P ) /\ X e. ( Base ` P ) ) -> ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) e. ( Base ` P ) )
21	3, 16, 18, 20	syl3anc 1326	. 2 |- ( ( R e. Ring /\ X e. C ) -> ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) e. ( Base ` P ) )
22		simpl 473	. . . . 5 |- ( ( R e. Ring /\ X e. C ) -> R e. Ring )
23		eqid 2622	. . . . . . . 8 |- (RLReg ` R ) = (RLReg ` R )
24	23, 10	unitrrg 19293	. . . . . . 7 |- ( R e. Ring -> (Unit ` R ) C_ (RLReg ` R ) )
25	24	adantr 481	. . . . . 6 |- ( ( R e. Ring /\ X e. C ) -> (Unit ` R ) C_ (RLReg ` R ) )
26	10, 13	unitinvcl 18674	. . . . . . 7 |- ( ( R e. Ring /\ ( (coe1 ` X ) ` ( D ` X ) ) e. (Unit ` R ) ) -> ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) e. (Unit ` R ) )
27	12, 26	sylan2 491	. . . . . 6 |- ( ( R e. Ring /\ X e. C ) -> ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) e. (Unit ` R ) )
28	25, 27	sseldd 3604	. . . . 5 |- ( ( R e. Ring /\ X e. C ) -> ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) e. (RLReg ` R ) )
29	9, 1, 23, 6, 19, 4	deg1mul3 23875	. . . . 5 |- ( ( R e. Ring /\ ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) e. (RLReg ` R ) /\ X e. ( Base ` P ) ) -> ( D ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) = ( D ` X ) )
30	22, 28, 18, 29	syl3anc 1326	. . . 4 |- ( ( R e. Ring /\ X e. C ) -> ( D ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) = ( D ` X ) )
31	9, 11	uc1pdeg 23907	. . . 4 |- ( ( R e. Ring /\ X e. C ) -> ( D ` X ) e. NN0 )
32	30, 31	eqeltrd 2701	. . 3 |- ( ( R e. Ring /\ X e. C ) -> ( D ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) e. NN0 )
33		eqid 2622	. . . . 5 |- ( 0g ` P ) = ( 0g ` P )
34	9, 1, 33, 6	deg1nn0clb 23850	. . . 4 |- ( ( R e. Ring /\ ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) e. ( Base ` P ) ) -> ( ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) =/= ( 0g ` P ) <-> ( D ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) e. NN0 ) )
35	21, 34	syldan 487	. . 3 |- ( ( R e. Ring /\ X e. C ) -> ( ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) =/= ( 0g ` P ) <-> ( D ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) e. NN0 ) )
36	32, 35	mpbird 247	. 2 |- ( ( R e. Ring /\ X e. C ) -> ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) =/= ( 0g ` P ) )
37	30	fveq2d 6195	. . 3 |- ( ( R e. Ring /\ X e. C ) -> ( (coe1 ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) ` ( D ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) ) = ( (coe1 ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) ` ( D ` X ) ) )
38		eqid 2622	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
39	1, 6, 5, 4, 19, 38	coe1sclmul 19652	. . . . 5 |- ( ( R e. Ring /\ ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) e. ( Base ` R ) /\ X e. ( Base ` P ) ) -> (coe1 ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) = ( ( NN0 X. { ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) } ) oF ( .r ` R ) (coe1 ` X ) ) )
40	22, 15, 18, 39	syl3anc 1326	. . . 4 |- ( ( R e. Ring /\ X e. C ) -> (coe1 ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) = ( ( NN0 X. { ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) } ) oF ( .r ` R ) (coe1 ` X ) ) )
41	40	fveq1d 6193	. . 3 |- ( ( R e. Ring /\ X e. C ) -> ( (coe1 ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) ` ( D ` X ) ) = ( ( ( NN0 X. { ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) } ) oF ( .r ` R ) (coe1 ` X ) ) ` ( D ` X ) ) )
42		nn0ex 11298	. . . . . . 7 |- NN0 e. _V
43	42	a1i 11	. . . . . 6 |- ( ( R e. Ring /\ X e. C ) -> NN0 e. _V )
44		fvexd 6203	. . . . . 6 |- ( ( R e. Ring /\ X e. C ) -> ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) e. _V )
45		eqid 2622	. . . . . . . 8 |- (coe1 ` X ) = (coe1 ` X )
46	45, 6, 1, 5	coe1f 19581	. . . . . . 7 |- ( X e. ( Base ` P ) -> (coe1 ` X ) : NN0 --> ( Base ` R ) )
47		ffn 6045	. . . . . . 7 |- ( (coe1 ` X ) : NN0 --> ( Base ` R ) -> (coe1 ` X ) Fn NN0 )
48	18, 46, 47	3syl 18	. . . . . 6 |- ( ( R e. Ring /\ X e. C ) -> (coe1 ` X ) Fn NN0 )
49		eqidd 2623	. . . . . 6 |- ( ( ( R e. Ring /\ X e. C ) /\ ( D ` X ) e. NN0 ) -> ( (coe1 ` X ) ` ( D ` X ) ) = ( (coe1 ` X ) ` ( D ` X ) ) )
50	43, 44, 48, 49	ofc1 6920	. . . . 5 |- ( ( ( R e. Ring /\ X e. C ) /\ ( D ` X ) e. NN0 ) -> ( ( ( NN0 X. { ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) } ) oF ( .r ` R ) (coe1 ` X ) ) ` ( D ` X ) ) = ( ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ( .r ` R ) ( (coe1 ` X ) ` ( D ` X ) ) ) )
51	31, 50	mpdan 702	. . . 4 |- ( ( R e. Ring /\ X e. C ) -> ( ( ( NN0 X. { ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) } ) oF ( .r ` R ) (coe1 ` X ) ) ` ( D ` X ) ) = ( ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ( .r ` R ) ( (coe1 ` X ) ` ( D ` X ) ) ) )
52		eqid 2622	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
53	10, 13, 38, 52	unitlinv 18677	. . . . 5 |- ( ( R e. Ring /\ ( (coe1 ` X ) ` ( D ` X ) ) e. (Unit ` R ) ) -> ( ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ( .r ` R ) ( (coe1 ` X ) ` ( D ` X ) ) ) = ( 1r ` R ) )
54	12, 53	sylan2 491	. . . 4 |- ( ( R e. Ring /\ X e. C ) -> ( ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ( .r ` R ) ( (coe1 ` X ) ` ( D ` X ) ) ) = ( 1r ` R ) )
55	51, 54	eqtrd 2656	. . 3 |- ( ( R e. Ring /\ X e. C ) -> ( ( ( NN0 X. { ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) } ) oF ( .r ` R ) (coe1 ` X ) ) ` ( D ` X ) ) = ( 1r ` R ) )
56	37, 41, 55	3eqtrd 2660	. 2 |- ( ( R e. Ring /\ X e. C ) -> ( (coe1 ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) ` ( D ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) ) = ( 1r ` R ) )
57		uc1pmon1p.m	. . 3 |- M = (Monic1p ` R )
58	1, 6, 33, 9, 57, 52	ismon1p 23902	. 2 |- ( ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) e. M <-> ( ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) e. ( Base ` P ) /\ ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) =/= ( 0g ` P ) /\ ( (coe1 ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) ` ( D ` ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) ) ) = ( 1r ` R ) ) )
59	21, 36, 56, 58	syl3anbrc 1246	1 |- ( ( R e. Ring /\ X e. C ) -> ( ( A ` ( I ` ( (coe1 ` X ) ` ( D ` X ) ) ) ) .x. X ) e. M )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   {csn 4177   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   NN0cn0 11292   Basecbs 15857   .rcmulr 15942   0gc0g 16100   1rcur 18501   Ringcrg 18547  Unitcui 18639   invrcinvr 18671  RLRegcrlreg 19279  algSccascl 19311  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814  Monic_1pcmn1 23885  Unic_1pcuc1p 23886

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-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iin 4523  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-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-subrg 18778  df-lmod 18865  df-lss 18933  df-rlreg 19283  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-mon1 23890  df-uc1p 23891

This theorem is referenced by:  ig1peu  23931