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Theorem ply1divalg2 26087

Description: Reverse the order of multiplication in ply1divalg 26086 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 = (0_g‘𝑃)
ply1divalg.t	⊢ ∙ = (._r‘𝑃)
ply1divalg.r1	⊢ (𝜑 → 𝑅 ∈ Ring)
ply1divalg.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
ply1divalg.g1	⊢ (𝜑 → 𝐺 ∈ 𝐵)
ply1divalg.g2	⊢ (𝜑 → 𝐺 ≠ 0 )
ply1divalg.g3	⊢ (𝜑 → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈)
ply1divalg.u	⊢ 𝑈 = (Unit‘𝑅)

**Assertion**
Ref	Expression
ply1divalg2	⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺))

Distinct variable groups: 𝜑,𝑞 𝐵,𝑞 𝐷,𝑞 𝐹,𝑞 𝐺,𝑞 − ,𝑞 𝑃,𝑞 𝑅,𝑞 ∙ ,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 2725	. . 3 ⊢ (Poly₁‘(opp_r‘𝑅)) = (Poly₁‘(opp_r‘𝑅))
2		ply1divalg.d	. . . 4 ⊢ 𝐷 = ( deg₁ ‘𝑅)
3		eqidd 2726	. . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘𝑅))
4		eqid 2725	. . . . . . . 8 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
5		eqid 2725	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	4, 5	opprbas 20279	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(opp_r‘𝑅))
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘(opp_r‘𝑅)))
8		eqid 2725	. . . . . . . . 9 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
9	4, 8	oppradd 20281	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘(opp_r‘𝑅))
10	9	oveqi 7426	. . . . . . 7 ⊢ (𝑞(+_g‘𝑅)𝑟) = (𝑞(+_g‘(opp_r‘𝑅))𝑟)
11	10	a1i 11	. . . . . 6 ⊢ ((⊤ ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞(+_g‘𝑅)𝑟) = (𝑞(+_g‘(opp_r‘𝑅))𝑟))
12	3, 7, 11	deg1propd 26035	. . . . 5 ⊢ (⊤ → ( deg₁ ‘𝑅) = ( deg₁ ‘(opp_r‘𝑅)))
13	12	mptru 1540	. . . 4 ⊢ ( deg₁ ‘𝑅) = ( deg₁ ‘(opp_r‘𝑅))
14	2, 13	eqtri 2753	. . 3 ⊢ 𝐷 = ( deg₁ ‘(opp_r‘𝑅))
15		ply1divalg.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
16		ply1divalg.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
17	16	fveq2i 6893	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(Poly₁‘𝑅))
18	3, 7, 11	ply1baspropd 22165	. . . . . 6 ⊢ (⊤ → (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘(opp_r‘𝑅))))
19	18	mptru 1540	. . . . 5 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘(opp_r‘𝑅)))
20	17, 19	eqtri 2753	. . . 4 ⊢ (Base‘𝑃) = (Base‘(Poly₁‘(opp_r‘𝑅)))
21	15, 20	eqtri 2753	. . 3 ⊢ 𝐵 = (Base‘(Poly₁‘(opp_r‘𝑅)))
22		ply1divalg.m	. . . 4 ⊢ − = (-_g‘𝑃)
23	20	a1i 11	. . . . . 6 ⊢ (⊤ → (Base‘𝑃) = (Base‘(Poly₁‘(opp_r‘𝑅))))
24	16	fveq2i 6893	. . . . . . . 8 ⊢ (+_g‘𝑃) = (+_g‘(Poly₁‘𝑅))
25	3, 7, 11	ply1plusgpropd 22166	. . . . . . . . 9 ⊢ (⊤ → (+_g‘(Poly₁‘𝑅)) = (+_g‘(Poly₁‘(opp_r‘𝑅))))
26	25	mptru 1540	. . . . . . . 8 ⊢ (+_g‘(Poly₁‘𝑅)) = (+_g‘(Poly₁‘(opp_r‘𝑅)))
27	24, 26	eqtri 2753	. . . . . . 7 ⊢ (+_g‘𝑃) = (+_g‘(Poly₁‘(opp_r‘𝑅)))
28	27	a1i 11	. . . . . 6 ⊢ (⊤ → (+_g‘𝑃) = (+_g‘(Poly₁‘(opp_r‘𝑅))))
29	23, 28	grpsubpropd 19000	. . . . 5 ⊢ (⊤ → (-_g‘𝑃) = (-_g‘(Poly₁‘(opp_r‘𝑅))))
30	29	mptru 1540	. . . 4 ⊢ (-_g‘𝑃) = (-_g‘(Poly₁‘(opp_r‘𝑅)))
31	22, 30	eqtri 2753	. . 3 ⊢ − = (-_g‘(Poly₁‘(opp_r‘𝑅)))
32		ply1divalg.z	. . . 4 ⊢ 0 = (0_g‘𝑃)
33	15	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐵 = (Base‘𝑃))
34	21	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐵 = (Base‘(Poly₁‘(opp_r‘𝑅))))
35	27	oveqi 7426	. . . . . . 7 ⊢ (𝑞(+_g‘𝑃)𝑟) = (𝑞(+_g‘(Poly₁‘(opp_r‘𝑅)))𝑟)
36	35	a1i 11	. . . . . 6 ⊢ ((⊤ ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+_g‘𝑃)𝑟) = (𝑞(+_g‘(Poly₁‘(opp_r‘𝑅)))𝑟))
37	33, 34, 36	grpidpropd 18616	. . . . 5 ⊢ (⊤ → (0_g‘𝑃) = (0_g‘(Poly₁‘(opp_r‘𝑅))))
38	37	mptru 1540	. . . 4 ⊢ (0_g‘𝑃) = (0_g‘(Poly₁‘(opp_r‘𝑅)))
39	32, 38	eqtri 2753	. . 3 ⊢ 0 = (0_g‘(Poly₁‘(opp_r‘𝑅)))
40		eqid 2725	. . 3 ⊢ (._r‘(Poly₁‘(opp_r‘𝑅))) = (._r‘(Poly₁‘(opp_r‘𝑅)))
41		ply1divalg.r1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
42	4	opprring 20285	. . . 4 ⊢ (𝑅 ∈ Ring → (opp_r‘𝑅) ∈ Ring)
43	41, 42	syl 17	. . 3 ⊢ (𝜑 → (opp_r‘𝑅) ∈ Ring)
44		ply1divalg.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
45		ply1divalg.g1	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
46		ply1divalg.g2	. . 3 ⊢ (𝜑 → 𝐺 ≠ 0 )
47		ply1divalg.g3	. . 3 ⊢ (𝜑 → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈)
48		ply1divalg.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
49	48, 4	opprunit 20315	. . 3 ⊢ 𝑈 = (Unit‘(opp_r‘𝑅))
50	1, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49	ply1divalg 26086	. 2 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺(._r‘(Poly₁‘(opp_r‘𝑅)))𝑞))) < (𝐷‘𝐺))
51	41	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑅 ∈ Ring)
52	45	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵)
53		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵)
54		ply1divalg.t	. . . . . . . . 9 ⊢ ∙ = (._r‘𝑃)
55	16, 4, 1, 54, 40, 15	ply1opprmul 22161	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺(._r‘(Poly₁‘(opp_r‘𝑅)))𝑞) = (𝑞 ∙ 𝐺))
56	51, 52, 53, 55	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐺(._r‘(Poly₁‘(opp_r‘𝑅)))𝑞) = (𝑞 ∙ 𝐺))
57	56	eqcomd 2731	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝑞 ∙ 𝐺) = (𝐺(._r‘(Poly₁‘(opp_r‘𝑅)))𝑞))
58	57	oveq2d 7429	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹 − (𝑞 ∙ 𝐺)) = (𝐹 − (𝐺(._r‘(Poly₁‘(opp_r‘𝑅)))𝑞)))
59	58	fveq2d 6894	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) = (𝐷‘(𝐹 − (𝐺(._r‘(Poly₁‘(opp_r‘𝑅)))𝑞))))
60	59	breq1d 5154	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺(._r‘(Poly₁‘(opp_r‘𝑅)))𝑞))) < (𝐷‘𝐺)))
61	60	reubidva 3380	. 2 ⊢ (𝜑 → (∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺) ↔ ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺(._r‘(Poly₁‘(opp_r‘𝑅)))𝑞))) < (𝐷‘𝐺)))
62	50, 61	mpbird 256	1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2930 ∃!wreu 3362 class class class wbr 5144 ‘cfv 6543 (class class class)co 7413 < clt 11273 Basecbs 17174 +_gcplusg 17227 ._rcmulr 17228 0_gc0g 17415 -_gcsg 18891 Ringcrg 20172 opp_rcoppr 20271 Unitcui 20293 Poly₁cpl1 22099 coe₁cco1 22100 deg₁ cdg1 26000

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-rlreg 21229 df-cnfld 21279 df-psr 21841 df-mvr 21842 df-mpl 21843 df-opsr 21845 df-psr1 22102 df-vr1 22103 df-ply1 22104 df-coe1 22105 df-mdeg 26001 df-deg1 26002

This theorem is referenced by: q1peqb 26104

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