evls1addd - Mathbox for Thierry Arnoux

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Theorem evls1addd 33081

Description: Univariate polynomial evaluation of a sum of polynomials. (Contributed by Thierry Arnoux, 8-Feb-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)
evls1addd.1	⊢ ⨣ = (+_g‘𝑊)
evls1addd.2	⊢ + = (+_g‘𝑆)
evls1addd.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evls1addd.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evls1addd.m	⊢ (𝜑 → 𝑀 ∈ 𝐵)
evls1addd.n	⊢ (𝜑 → 𝑁 ∈ 𝐵)
evls1addd.y	⊢ (𝜑 → 𝐶 ∈ 𝐾)

**Assertion**
Ref	Expression
evls1addd	⊢ (𝜑 → ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) + ((𝑄‘𝑁)‘𝐶)))

**Proof of Theorem evls1addd**
Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝜑 → 𝜑)
2		evls1addd.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
3		evls1addd.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
4		eqid 2724	. . . . . . 7 ⊢ (Poly₁‘𝑆) = (Poly₁‘𝑆)
5		ressply1evl.u	. . . . . . 7 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
6		ressply1evl.w	. . . . . . 7 ⊢ 𝑊 = (Poly₁‘𝑈)
7		ressply1evl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
8		evls1addd.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
9		eqid 2724	. . . . . . 7 ⊢ ((Poly₁‘𝑆) ↾_s 𝐵) = ((Poly₁‘𝑆) ↾_s 𝐵)
10	4, 5, 6, 7, 8, 9	ressply1add 22070	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵)) → (𝑀(+_g‘𝑊)𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
11	1, 2, 3, 10	syl12anc 834	. . . . 5 ⊢ (𝜑 → (𝑀(+_g‘𝑊)𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
12		evls1addd.1	. . . . . 6 ⊢ ⨣ = (+_g‘𝑊)
13	12	oveqi 7414	. . . . 5 ⊢ (𝑀 ⨣ 𝑁) = (𝑀(+_g‘𝑊)𝑁)
14	7	fvexi 6895	. . . . . . 7 ⊢ 𝐵 ∈ V
15		eqid 2724	. . . . . . . 8 ⊢ (+_g‘(Poly₁‘𝑆)) = (+_g‘(Poly₁‘𝑆))
16	9, 15	ressplusg 17233	. . . . . . 7 ⊢ (𝐵 ∈ V → (+_g‘(Poly₁‘𝑆)) = (+_g‘((Poly₁‘𝑆) ↾_s 𝐵)))
17	14, 16	ax-mp 5	. . . . . 6 ⊢ (+_g‘(Poly₁‘𝑆)) = (+_g‘((Poly₁‘𝑆) ↾_s 𝐵))
18	17	oveqi 7414	. . . . 5 ⊢ (𝑀(+_g‘(Poly₁‘𝑆))𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁)
19	11, 13, 18	3eqtr4g 2789	. . . 4 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) = (𝑀(+_g‘(Poly₁‘𝑆))𝑁))
20	19	fveq2d 6885	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁)))
21	20	fveq1d 6883	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁))‘𝐶) = (((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁))‘𝐶))
22		ressply1evl.q	. . . . . 6 ⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
23		ressply1evl.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
24		eqid 2724	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
25		evls1addd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
26	22, 23, 6, 5, 7, 24, 25, 8	ressply1evl 33080	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
27	26	fveq1d 6883	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ⨣ 𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)))
28	5	subrgring 20465	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
29	6	ply1ring 22088	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
30	8, 28, 29	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
31	30	ringgrpd 20136	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
32	7, 12, 31, 2, 3	grpcld 18866	. . . . 5 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) ∈ 𝐵)
33	32	fvresd 6901	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)))
34	27, 33	eqtr2d 2765	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = (𝑄‘(𝑀 ⨣ 𝑁)))
35	34	fveq1d 6883	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁))‘𝐶) = ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶))
36		eqid 2724	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
37		evls1addd.y	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
38		eqid 2724	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
39		eqid 2724	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
40	4, 5, 6, 7, 8, 38, 39, 36	ressply1bas2 22068	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
41		inss2 4221	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
42	40, 41	eqsstrdi 4028	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
43	42, 2	sseldd 3975	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
44	26	fveq1d 6883	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
45	2	fvresd 6901	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
46	44, 45	eqtr2d 2765	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
47	46	fveq1d 6883	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶))
48	43, 47	jca 511	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶)))
49	42, 3	sseldd 3975	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
50	26	fveq1d 6883	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
51	3	fvresd 6901	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
52	50, 51	eqtr2d 2765	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
53	52	fveq1d 6883	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶))
54	49, 53	jca 511	. . . 4 ⊢ (𝜑 → (𝑁 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶)))
55		evls1addd.2	. . . 4 ⊢ + = (+_g‘𝑆)
56	24, 4, 23, 36, 25, 37, 48, 54, 15, 55	evl1addd 22181	. . 3 ⊢ (𝜑 → ((𝑀(+_g‘(Poly₁‘𝑆))𝑁) ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) + ((𝑄‘𝑁)‘𝐶))))
57	56	simprd 495	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) + ((𝑄‘𝑁)‘𝐶)))
58	21, 35, 57	3eqtr3d 2772	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 3466 ∩ cin 3939 ↾ cres 5668 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 ↾_s cress 17171 +_gcplusg 17195 Ringcrg 20127 CRingccrg 20128 SubRingcsubrg 20458 PwSer₁cps1 22016 Poly₁cpl1 22018 evalSub₁ ces1 22153 eval₁ce1 22154

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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-hom 17219 df-cco 17220 df-0g 17385 df-gsum 17386 df-prds 17391 df-pws 17393 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-mhm 18702 df-submnd 18703 df-grp 18855 df-minusg 18856 df-sbg 18857 df-mulg 18985 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-srg 20081 df-ring 20129 df-cring 20130 df-rhm 20363 df-subrng 20435 df-subrg 20460 df-lmod 20697 df-lss 20768 df-lsp 20808 df-assa 21715 df-asp 21716 df-ascl 21717 df-psr 21770 df-mvr 21771 df-mpl 21772 df-opsr 21774 df-evls 21944 df-evl 21945 df-psr1 22021 df-vr1 22022 df-ply1 22023 df-coe1 22024 df-evls1 22155 df-evl1 22156

This theorem is referenced by: evls1maprhm 33205

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