Theorem ply1divalg2 23898

Description: Reverse the order of multiplication in ply1divalg 23897 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
ply1divalg.p	|- P = (Poly1 ` R )
ply1divalg.d	|- D = ( deg1 ` R )
ply1divalg.b	|- B = ( Base ` P )
ply1divalg.m	|- .- = ( -g ` P )
ply1divalg.z	|- .0. = ( 0g ` P )
ply1divalg.t	|- .xb = ( .r ` P )
ply1divalg.r1	|- ( ph -> R e. Ring )
ply1divalg.f	|- ( ph -> F e. B )
ply1divalg.g1	|- ( ph -> G e. B )
ply1divalg.g2	|- ( ph -> G =/= .0. )
ply1divalg.g3	|- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) e. U )
ply1divalg.u	|- U = (Unit ` R )

Assertion

Ref	Expression
ply1divalg2	|- ( ph -> E! q e. B ( D ` ( F .- ( q .xb G ) ) ) < ( D ` G ) )

Distinct variable groups:   ph, q   B, q   D, q   F, q   G, q   .- , q   P, q   R, q   .xb , q   .0. , q

Allowed substitution hint:   U( q)

Proof of Theorem ply1divalg2

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (Poly1 ` (oppr ` R ) ) = (Poly1 ` (oppr ` R ) )
2		ply1divalg.d	. . . 4 |- D = ( deg1 ` R )
3		eqidd 2623	. . . . . 6 |- ( T. -> ( Base ` R ) = ( Base ` R ) )
4		eqid 2622	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
5		eqid 2622	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
6	4, 5	opprbas 18629	. . . . . . 7 |- ( Base ` R ) = ( Base ` (oppr ` R ) )
7	6	a1i 11	. . . . . 6 |- ( T. -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
8		eqid 2622	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
9	4, 8	oppradd 18630	. . . . . . . 8 |- ( +g ` R ) = ( +g ` (oppr ` R ) )
10	9	oveqi 6663	. . . . . . 7 |- ( q ( +g ` R ) r ) = ( q ( +g ` (oppr ` R ) ) r )
11	10	a1i 11	. . . . . 6 |- ( ( T. /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( +g ` R ) r ) = ( q ( +g ` (oppr ` R ) ) r ) )
12	3, 7, 11	deg1propd 23846	. . . . 5 |- ( T. -> ( deg1 ` R ) = ( deg1 ` (oppr ` R ) ) )
13	12	trud 1493	. . . 4 |- ( deg1 ` R ) = ( deg1 ` (oppr ` R ) )
14	2, 13	eqtri 2644	. . 3 |- D = ( deg1 ` (oppr ` R ) )
15		ply1divalg.b	. . . 4 |- B = ( Base ` P )
16		ply1divalg.p	. . . . . 6 |- P = (Poly1 ` R )
17	16	fveq2i 6194	. . . . 5 |- ( Base ` P ) = ( Base ` (Poly1 ` R ) )
18	3, 7, 11	ply1baspropd 19613	. . . . . 6 |- ( T. -> ( Base ` (Poly1 ` R ) ) = ( Base ` (Poly1 ` (oppr ` R ) ) ) )
19	18	trud 1493	. . . . 5 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (Poly1 ` (oppr ` R ) ) )
20	17, 19	eqtri 2644	. . . 4 |- ( Base ` P ) = ( Base ` (Poly1 ` (oppr ` R ) ) )
21	15, 20	eqtri 2644	. . 3 |- B = ( Base ` (Poly1 ` (oppr ` R ) ) )
22		ply1divalg.m	. . . 4 |- .- = ( -g ` P )
23	20	a1i 11	. . . . . 6 |- ( T. -> ( Base ` P ) = ( Base ` (Poly1 ` (oppr ` R ) ) ) )
24	16	fveq2i 6194	. . . . . . . 8 |- ( +g ` P ) = ( +g ` (Poly1 ` R ) )
25	3, 7, 11	ply1plusgpropd 19614	. . . . . . . . 9 |- ( T. -> ( +g ` (Poly1 ` R ) ) = ( +g ` (Poly1 ` (oppr ` R ) ) ) )
26	25	trud 1493	. . . . . . . 8 |- ( +g ` (Poly1 ` R ) ) = ( +g ` (Poly1 ` (oppr ` R ) ) )
27	24, 26	eqtri 2644	. . . . . . 7 |- ( +g ` P ) = ( +g ` (Poly1 ` (oppr ` R ) ) )
28	27	a1i 11	. . . . . 6 |- ( T. -> ( +g ` P ) = ( +g ` (Poly1 ` (oppr ` R ) ) ) )
29	23, 28	grpsubpropd 17520	. . . . 5 |- ( T. -> ( -g ` P ) = ( -g ` (Poly1 ` (oppr ` R ) ) ) )
30	29	trud 1493	. . . 4 |- ( -g ` P ) = ( -g ` (Poly1 ` (oppr ` R ) ) )
31	22, 30	eqtri 2644	. . 3 |- .- = ( -g ` (Poly1 ` (oppr ` R ) ) )
32		ply1divalg.z	. . . 4 |- .0. = ( 0g ` P )
33	15	a1i 11	. . . . . 6 |- ( T. -> B = ( Base ` P ) )
34	21	a1i 11	. . . . . 6 |- ( T. -> B = ( Base ` (Poly1 ` (oppr ` R ) ) ) )
35	27	oveqi 6663	. . . . . . 7 |- ( q ( +g ` P ) r ) = ( q ( +g ` (Poly1 ` (oppr ` R ) ) ) r )
36	35	a1i 11	. . . . . 6 |- ( ( T. /\ ( q e. B /\ r e. B ) ) -> ( q ( +g ` P ) r ) = ( q ( +g ` (Poly1 ` (oppr ` R ) ) ) r ) )
37	33, 34, 36	grpidpropd 17261	. . . . 5 |- ( T. -> ( 0g ` P ) = ( 0g ` (Poly1 ` (oppr ` R ) ) ) )
38	37	trud 1493	. . . 4 |- ( 0g ` P ) = ( 0g ` (Poly1 ` (oppr ` R ) ) )
39	32, 38	eqtri 2644	. . 3 |- .0. = ( 0g ` (Poly1 ` (oppr ` R ) ) )
40		eqid 2622	. . 3 |- ( .r ` (Poly1 ` (oppr ` R ) ) ) = ( .r ` (Poly1 ` (oppr ` R ) ) )
41		ply1divalg.r1	. . . 4 |- ( ph -> R e. Ring )
42	4	opprring 18631	. . . 4 |- ( R e. Ring -> (oppr ` R ) e. Ring )
43	41, 42	syl 17	. . 3 |- ( ph -> (oppr ` R ) e. Ring )
44		ply1divalg.f	. . 3 |- ( ph -> F e. B )
45		ply1divalg.g1	. . 3 |- ( ph -> G e. B )
46		ply1divalg.g2	. . 3 |- ( ph -> G =/= .0. )
47		ply1divalg.g3	. . 3 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) e. U )
48		ply1divalg.u	. . . 4 |- U = (Unit ` R )
49	48, 4	opprunit 18661	. . 3 |- U = (Unit ` (oppr ` R ) )
50	1, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49	ply1divalg 23897	. 2 |- ( ph -> E! q e. B ( D ` ( F .- ( G ( .r ` (Poly1 ` (oppr ` R ) ) ) q ) ) ) < ( D ` G ) )
51	41	adantr 481	. . . . . . . 8 |- ( ( ph /\ q e. B ) -> R e. Ring )
52	45	adantr 481	. . . . . . . 8 |- ( ( ph /\ q e. B ) -> G e. B )
53		simpr 477	. . . . . . . 8 |- ( ( ph /\ q e. B ) -> q e. B )
54		ply1divalg.t	. . . . . . . . 9 |- .xb = ( .r ` P )
55	16, 4, 1, 54, 40, 15	ply1opprmul 19609	. . . . . . . 8 |- ( ( R e. Ring /\ G e. B /\ q e. B ) -> ( G ( .r ` (Poly1 ` (oppr ` R ) ) ) q ) = ( q .xb G ) )
56	51, 52, 53, 55	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ q e. B ) -> ( G ( .r ` (Poly1 ` (oppr ` R ) ) ) q ) = ( q .xb G ) )
57	56	eqcomd 2628	. . . . . 6 |- ( ( ph /\ q e. B ) -> ( q .xb G ) = ( G ( .r ` (Poly1 ` (oppr ` R ) ) ) q ) )
58	57	oveq2d 6666	. . . . 5 |- ( ( ph /\ q e. B ) -> ( F .- ( q .xb G ) ) = ( F .- ( G ( .r ` (Poly1 ` (oppr ` R ) ) ) q ) ) )
59	58	fveq2d 6195	. . . 4 |- ( ( ph /\ q e. B ) -> ( D ` ( F .- ( q .xb G ) ) ) = ( D ` ( F .- ( G ( .r ` (Poly1 ` (oppr ` R ) ) ) q ) ) ) )
60	59	breq1d 4663	. . 3 |- ( ( ph /\ q e. B ) -> ( ( D ` ( F .- ( q .xb G ) ) ) < ( D ` G ) <-> ( D ` ( F .- ( G ( .r ` (Poly1 ` (oppr ` R ) ) ) q ) ) ) < ( D ` G ) ) )
61	60	reubidva 3125	. 2 |- ( ph -> ( E! q e. B ( D ` ( F .- ( q .xb G ) ) ) < ( D ` G ) <-> E! q e. B ( D ` ( F .- ( G ( .r ` (Poly1 ` (oppr ` R ) ) ) q ) ) ) < ( D ` G ) ) )
62	50, 61	mpbird 247	1 |- ( ph -> E! q e. B ( D ` ( F .- ( q .xb G ) ) ) < ( D ` G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   E!wreu 2914   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   < clt 10074   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   -gcsg 17424   Ringcrg 18547  opp_rcoppr 18622  Unitcui 18639  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814

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  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-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

This theorem is referenced by:  q1peqb  23914