evls1muld - Metamath Proof Explorer

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Theorem evls1muld 22376

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 2737	. . . . . . 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 2737	. . . . . . 7 ⊢ ((Poly₁‘𝑆) ↾_s 𝐵) = ((Poly₁‘𝑆) ↾_s 𝐵)
10	4, 5, 6, 7, 8, 9	ressply1mul 22232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵)) → (𝑀(._r‘𝑊)𝑁) = (𝑀(._r‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
11	1, 2, 3, 10	syl12anc 837	. . . . 5 ⊢ (𝜑 → (𝑀(._r‘𝑊)𝑁) = (𝑀(._r‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
12		evls1muld.1	. . . . . 6 ⊢ × = (._r‘𝑊)
13	12	oveqi 7444	. . . . 5 ⊢ (𝑀 × 𝑁) = (𝑀(._r‘𝑊)𝑁)
14	7	fvexi 6920	. . . . . . 7 ⊢ 𝐵 ∈ V
15		eqid 2737	. . . . . . . 8 ⊢ (._r‘(Poly₁‘𝑆)) = (._r‘(Poly₁‘𝑆))
16	9, 15	ressmulr 17351	. . . . . . 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 7444	. . . . 5 ⊢ (𝑀(._r‘(Poly₁‘𝑆))𝑁) = (𝑀(._r‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁)
19	11, 13, 18	3eqtr4g 2802	. . . 4 ⊢ (𝜑 → (𝑀 × 𝑁) = (𝑀(._r‘(Poly₁‘𝑆))𝑁))
20	19	fveq2d 6910	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 × 𝑁)) = ((eval₁‘𝑆)‘(𝑀(._r‘(Poly₁‘𝑆))𝑁)))
21	20	fveq1d 6908	. 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 2737	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
25		evls1muld.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
26	22, 23, 6, 5, 7, 24, 25, 8	ressply1evl 22374	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
27	26	fveq1d 6908	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 × 𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 × 𝑁)))
28	5	subrgring 20574	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
29	6	ply1ring 22249	. . . . . . 7 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
30	8, 28, 29	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Ring)
31	7, 12, 30, 2, 3	ringcld 20257	. . . . 5 ⊢ (𝜑 → (𝑀 × 𝑁) ∈ 𝐵)
32	31	fvresd 6926	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 × 𝑁)) = ((eval₁‘𝑆)‘(𝑀 × 𝑁)))
33	27, 32	eqtr2d 2778	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 × 𝑁)) = (𝑄‘(𝑀 × 𝑁)))
34	33	fveq1d 6908	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 × 𝑁))‘𝐶) = ((𝑄‘(𝑀 × 𝑁))‘𝐶))
35		eqid 2737	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
36		evls1muld.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
37		eqid 2737	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
38		eqid 2737	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
39	4, 5, 6, 7, 8, 37, 38, 35	ressply1bas2 22229	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
40		inss2 4238	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
41	39, 40	eqsstrdi 4028	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
42	41, 2	sseldd 3984	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
43	26	fveq1d 6908	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
44	2	fvresd 6926	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
45	43, 44	eqtr2d 2778	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
46	45	fveq1d 6908	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶))
47	42, 46	jca 511	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶)))
48	41, 3	sseldd 3984	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
49	26	fveq1d 6908	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
50	3	fvresd 6926	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
51	49, 50	eqtr2d 2778	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
52	51	fveq1d 6908	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶))
53	48, 52	jca 511	. . . 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 22347	. . 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 495	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀(._r‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))
57	21, 34, 56	3eqtr3d 2785	1 ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾_s cress 17274 ._rcmulr 17298 Ringcrg 20230 CRingccrg 20231 SubRingcsubrg 20569 PwSer₁cps1 22176 Poly₁cpl1 22178 evalSub₁ ces1 22317 eval₁ce1 22318

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-evls1 22319 df-evl1 22320

This theorem is referenced by: evls1maprhm 22380 irngnzply1lem 33740 minplyirred 33754 irredminply 33757

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