evls1muld - Metamath Proof Explorer

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Theorem evls1muld 22298

Description: Univariate polynomial evaluation of a product of polynomials. (Contributed by Thierry Arnoux, 24-Jan-2025.)

**Hypotheses**
Ref	Expression
ressply1evl2.q	⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
ressply1evl2.k	⊢ 𝐾 = (Base‘𝑆)
ressply1evl2.w	⊢ 𝑊 = (Poly₁‘𝑈)
ressply1evl2.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
ressply1evl2.b	⊢ 𝐵 = (Base‘𝑊)
evls1muld.1	⊢ × = (._r‘𝑊)
evls1muld.2	⊢ · = (._r‘𝑆)
evls1muld.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evls1muld.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evls1muld.m	⊢ (𝜑 → 𝑀 ∈ 𝐵)
evls1muld.n	⊢ (𝜑 → 𝑁 ∈ 𝐵)
evls1muld.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)

**Assertion**
Ref	Expression
evls1muld	⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))

**Proof of Theorem evls1muld**
Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝜑 → 𝜑)
2		evls1muld.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
3		evls1muld.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
4		eqid 2725	. . . . . . 7 ⊢ (Poly₁‘𝑆) = (Poly₁‘𝑆)
5		ressply1evl2.u	. . . . . . 7 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
6		ressply1evl2.w	. . . . . . 7 ⊢ 𝑊 = (Poly₁‘𝑈)
7		ressply1evl2.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
8		evls1muld.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
9		eqid 2725	. . . . . . 7 ⊢ ((Poly₁‘𝑆) ↾_s 𝐵) = ((Poly₁‘𝑆) ↾_s 𝐵)
10	4, 5, 6, 7, 8, 9	ressply1mul 22156	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵)) → (𝑀(._r‘𝑊)𝑁) = (𝑀(._r‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
11	1, 2, 3, 10	syl12anc 835	. . . . 5 ⊢ (𝜑 → (𝑀(._r‘𝑊)𝑁) = (𝑀(._r‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
12		evls1muld.1	. . . . . 6 ⊢ × = (._r‘𝑊)
13	12	oveqi 7427	. . . . 5 ⊢ (𝑀 × 𝑁) = (𝑀(._r‘𝑊)𝑁)
14	7	fvexi 6904	. . . . . . 7 ⊢ 𝐵 ∈ V
15		eqid 2725	. . . . . . . 8 ⊢ (._r‘(Poly₁‘𝑆)) = (._r‘(Poly₁‘𝑆))
16	9, 15	ressmulr 17285	. . . . . . 7 ⊢ (𝐵 ∈ V → (._r‘(Poly₁‘𝑆)) = (._r‘((Poly₁‘𝑆) ↾_s 𝐵)))
17	14, 16	ax-mp 5	. . . . . 6 ⊢ (._r‘(Poly₁‘𝑆)) = (._r‘((Poly₁‘𝑆) ↾_s 𝐵))
18	17	oveqi 7427	. . . . 5 ⊢ (𝑀(._r‘(Poly₁‘𝑆))𝑁) = (𝑀(._r‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁)
19	11, 13, 18	3eqtr4g 2790	. . . 4 ⊢ (𝜑 → (𝑀 × 𝑁) = (𝑀(._r‘(Poly₁‘𝑆))𝑁))
20	19	fveq2d 6894	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 × 𝑁)) = ((eval₁‘𝑆)‘(𝑀(._r‘(Poly₁‘𝑆))𝑁)))
21	20	fveq1d 6892	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 × 𝑁))‘𝐶) = (((eval₁‘𝑆)‘(𝑀(._r‘(Poly₁‘𝑆))𝑁))‘𝐶))
22		ressply1evl2.q	. . . . . 6 ⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
23		ressply1evl2.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
24		eqid 2725	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
25		evls1muld.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
26	22, 23, 6, 5, 7, 24, 25, 8	ressply1evl 22296	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
27	26	fveq1d 6892	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 × 𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 × 𝑁)))
28	5	subrgring 20515	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
29	6	ply1ring 22173	. . . . . . 7 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
30	8, 28, 29	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Ring)
31	7, 12, 30, 2, 3	ringcld 20201	. . . . 5 ⊢ (𝜑 → (𝑀 × 𝑁) ∈ 𝐵)
32	31	fvresd 6910	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 × 𝑁)) = ((eval₁‘𝑆)‘(𝑀 × 𝑁)))
33	27, 32	eqtr2d 2766	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 × 𝑁)) = (𝑄‘(𝑀 × 𝑁)))
34	33	fveq1d 6892	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 × 𝑁))‘𝐶) = ((𝑄‘(𝑀 × 𝑁))‘𝐶))
35		eqid 2725	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
36		evls1muld.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
37		eqid 2725	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
38		eqid 2725	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
39	4, 5, 6, 7, 8, 37, 38, 35	ressply1bas2 22153	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
40		inss2 4222	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
41	39, 40	eqsstrdi 4026	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
42	41, 2	sseldd 3973	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
43	26	fveq1d 6892	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
44	2	fvresd 6910	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
45	43, 44	eqtr2d 2766	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
46	45	fveq1d 6892	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶))
47	42, 46	jca 510	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶)))
48	41, 3	sseldd 3973	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
49	26	fveq1d 6892	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
50	3	fvresd 6910	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
51	49, 50	eqtr2d 2766	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
52	51	fveq1d 6892	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶))
53	48, 52	jca 510	. . . 4 ⊢ (𝜑 → (𝑁 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶)))
54		evls1muld.2	. . . 4 ⊢ · = (._r‘𝑆)
55	24, 4, 23, 35, 25, 36, 47, 53, 15, 54	evl1muld 22269	. . 3 ⊢ (𝜑 → ((𝑀(._r‘(Poly₁‘𝑆))𝑁) ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘(𝑀(._r‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶))))
56	55	simprd 494	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀(._r‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))
57	21, 34, 56	3eqtr3d 2773	1 ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∩ cin 3938 ↾ cres 5672 ‘cfv 6541 (class class class)co 7414 Basecbs 17177 ↾_s cress 17206 ._rcmulr 17231 Ringcrg 20175 CRingccrg 20176 SubRingcsubrg 20508 PwSer₁cps1 22100 Poly₁cpl1 22102 evalSub₁ ces1 22239 eval₁ce1 22240

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 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-ofr 7681 df-om 7867 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 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 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-srg 20129 df-ring 20177 df-cring 20178 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-assa 21789 df-asp 21790 df-ascl 21791 df-psr 21844 df-mvr 21845 df-mpl 21846 df-opsr 21848 df-evls 22023 df-evl 22024 df-psr1 22105 df-vr1 22106 df-ply1 22107 df-coe1 22108 df-evls1 22241 df-evl1 22242

This theorem is referenced by: evls1maprhm 22302 irngnzply1lem 33397 minplyirred 33410 irredminply 33413

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