evls1subd - Mathbox for Thierry Arnoux

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Theorem evls1subd 33585

Description: Univariate polynomial evaluation of a difference of polynomials. (Contributed by Thierry Arnoux, 25-Apr-2025.)

**Hypotheses**
Ref	Expression
ressply1evl.q	⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
ressply1evl.k	⊢ 𝐾 = (Base‘𝑆)
ressply1evl.w	⊢ 𝑊 = (Poly₁‘𝑈)
ressply1evl.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
ressply1evl.b	⊢ 𝐵 = (Base‘𝑊)
evls1subd.1	⊢ 𝐷 = (-_g‘𝑊)
evls1subd.2	⊢ − = (-_g‘𝑆)
evls1subd.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evls1subd.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evls1subd.m	⊢ (𝜑 → 𝑀 ∈ 𝐵)
evls1subd.n	⊢ (𝜑 → 𝑁 ∈ 𝐵)
evls1subd.y	⊢ (𝜑 → 𝐶 ∈ 𝐾)

**Assertion**
Ref	Expression
evls1subd	⊢ (𝜑 → ((𝑄‘(𝑀𝐷𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))

**Proof of Theorem evls1subd**
Step	Hyp	Ref	Expression
1		evls1subd.1	. . . . . . 7 ⊢ 𝐷 = (-_g‘𝑊)
2	1	oveqi 7418	. . . . . 6 ⊢ (𝑀𝐷𝑁) = (𝑀(-_g‘𝑊)𝑁)
3		eqid 2735	. . . . . . 7 ⊢ (Poly₁‘𝑆) = (Poly₁‘𝑆)
4		ressply1evl.u	. . . . . . 7 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
5		ressply1evl.w	. . . . . . 7 ⊢ 𝑊 = (Poly₁‘𝑈)
6		ressply1evl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
7		evls1subd.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
8		eqid 2735	. . . . . . 7 ⊢ ((Poly₁‘𝑆) ↾_s 𝐵) = ((Poly₁‘𝑆) ↾_s 𝐵)
9		evls1subd.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
10		evls1subd.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
11	3, 4, 5, 6, 7, 8, 9, 10	ressply1sub 33583	. . . . . 6 ⊢ (𝜑 → (𝑀(-_g‘𝑊)𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
12	2, 11	eqtrid 2782	. . . . 5 ⊢ (𝜑 → (𝑀𝐷𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
13	3, 4, 5, 6	subrgply1 22168	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubRing‘(Poly₁‘𝑆)))
14		subrgsubg 20537	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘(Poly₁‘𝑆)) → 𝐵 ∈ (SubGrp‘(Poly₁‘𝑆)))
15	7, 13, 14	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘(Poly₁‘𝑆)))
16		eqid 2735	. . . . . . 7 ⊢ (-_g‘(Poly₁‘𝑆)) = (-_g‘(Poly₁‘𝑆))
17		eqid 2735	. . . . . . 7 ⊢ (-_g‘((Poly₁‘𝑆) ↾_s 𝐵)) = (-_g‘((Poly₁‘𝑆) ↾_s 𝐵))
18	16, 8, 17	subgsub 19121	. . . . . 6 ⊢ ((𝐵 ∈ (SubGrp‘(Poly₁‘𝑆)) ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀(-_g‘(Poly₁‘𝑆))𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
19	15, 9, 10, 18	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑀(-_g‘(Poly₁‘𝑆))𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
20	12, 19	eqtr4d 2773	. . . 4 ⊢ (𝜑 → (𝑀𝐷𝑁) = (𝑀(-_g‘(Poly₁‘𝑆))𝑁))
21	20	fveq2d 6880	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀𝐷𝑁)) = ((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁)))
22	21	fveq1d 6878	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀𝐷𝑁))‘𝐶) = (((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁))‘𝐶))
23		ressply1evl.q	. . . . . 6 ⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
24		ressply1evl.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
25		eqid 2735	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
26		evls1subd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
27	23, 24, 5, 4, 6, 25, 26, 7	ressply1evl 22308	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
28	27	fveq1d 6878	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀𝐷𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀𝐷𝑁)))
29	4	subrgring 20534	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
30	5	ply1ring 22183	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
31	7, 29, 30	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
32	31	ringgrpd 20202	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
33	6, 1	grpsubcl 19003	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀𝐷𝑁) ∈ 𝐵)
34	32, 9, 10, 33	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑀𝐷𝑁) ∈ 𝐵)
35	34	fvresd 6896	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀𝐷𝑁)) = ((eval₁‘𝑆)‘(𝑀𝐷𝑁)))
36	28, 35	eqtr2d 2771	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀𝐷𝑁)) = (𝑄‘(𝑀𝐷𝑁)))
37	36	fveq1d 6878	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀𝐷𝑁))‘𝐶) = ((𝑄‘(𝑀𝐷𝑁))‘𝐶))
38		eqid 2735	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
39		evls1subd.y	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
40		eqid 2735	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
41		eqid 2735	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
42	3, 4, 5, 6, 7, 40, 41, 38	ressply1bas2 22163	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
43		inss2 4213	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
44	42, 43	eqsstrdi 4003	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
45	44, 9	sseldd 3959	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
46	27	fveq1d 6878	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
47	9	fvresd 6896	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
48	46, 47	eqtr2d 2771	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
49	48	fveq1d 6878	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶))
50	45, 49	jca 511	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶)))
51	44, 10	sseldd 3959	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
52	27	fveq1d 6878	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
53	10	fvresd 6896	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
54	52, 53	eqtr2d 2771	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
55	54	fveq1d 6878	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶))
56	51, 55	jca 511	. . . 4 ⊢ (𝜑 → (𝑁 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶)))
57		evls1subd.2	. . . 4 ⊢ − = (-_g‘𝑆)
58	25, 3, 24, 38, 26, 39, 50, 56, 16, 57	evl1subd 22280	. . 3 ⊢ (𝜑 → ((𝑀(-_g‘(Poly₁‘𝑆))𝑁) ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶))))
59	58	simprd 495	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))
60	22, 37, 59	3eqtr3d 2778	1 ⊢ (𝜑 → ((𝑄‘(𝑀𝐷𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3925 ↾ cres 5656 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 ↾_s cress 17251 Grpcgrp 18916 -_gcsg 18918 SubGrpcsubg 19103 Ringcrg 20193 CRingccrg 20194 SubRingcsubrg 20529 PwSer₁cps1 22110 Poly₁cpl1 22112 evalSub₁ ces1 22251 eval₁ce1 22252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-ofr 7672 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-fz 13525 df-fzo 13672 df-seq 14020 df-hash 14349 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-hom 17295 df-cco 17296 df-0g 17455 df-gsum 17456 df-prds 17461 df-pws 17463 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-mulg 19051 df-subg 19106 df-ghm 19196 df-cntz 19300 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-srg 20147 df-ring 20195 df-cring 20196 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-lmod 20819 df-lss 20889 df-lsp 20929 df-assa 21813 df-asp 21814 df-ascl 21815 df-psr 21869 df-mvr 21870 df-mpl 21871 df-opsr 21873 df-evls 22032 df-evl 22033 df-psr1 22115 df-vr1 22116 df-ply1 22117 df-coe1 22118 df-evls1 22253 df-evl1 22254

This theorem is referenced by: irredminply 33750 2sqr3minply 33814

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