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Theorem evls1addd 22258

Description: Univariate polynomial evaluation of a sum of polynomials. (Contributed by Thierry Arnoux, 8-Feb-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)
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 2729	. . . . . . 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		evls1addd.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
9		eqid 2729	. . . . . . 7 ⊢ ((Poly₁‘𝑆) ↾_s 𝐵) = ((Poly₁‘𝑆) ↾_s 𝐵)
10	4, 5, 6, 7, 8, 9	ressply1add 22114	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵)) → (𝑀(+_g‘𝑊)𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
11	1, 2, 3, 10	syl12anc 836	. . . . 5 ⊢ (𝜑 → (𝑀(+_g‘𝑊)𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
12		evls1addd.1	. . . . . 6 ⊢ ⨣ = (+_g‘𝑊)
13	12	oveqi 7400	. . . . 5 ⊢ (𝑀 ⨣ 𝑁) = (𝑀(+_g‘𝑊)𝑁)
14	7	fvexi 6872	. . . . . . 7 ⊢ 𝐵 ∈ V
15		eqid 2729	. . . . . . . 8 ⊢ (+_g‘(Poly₁‘𝑆)) = (+_g‘(Poly₁‘𝑆))
16	9, 15	ressplusg 17254	. . . . . . 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 7400	. . . . 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 6862	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁)))
21	20	fveq1d 6860	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁))‘𝐶) = (((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁))‘𝐶))
22		ressply1evl2.q	. . . . . 6 ⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
23		ressply1evl2.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
24		eqid 2729	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
25		evls1addd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
26	22, 23, 6, 5, 7, 24, 25, 8	ressply1evl 22257	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
27	26	fveq1d 6860	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ⨣ 𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)))
28	5	subrgring 20483	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
29	6	ply1ring 22132	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
30	8, 28, 29	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
31	30	ringgrpd 20151	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
32	7, 12, 31, 2, 3	grpcld 18879	. . . . 5 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) ∈ 𝐵)
33	32	fvresd 6878	. . . 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 6860	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁))‘𝐶) = ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶))
36		eqid 2729	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
37		evls1addd.y	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
38		eqid 2729	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
39		eqid 2729	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
40	4, 5, 6, 7, 8, 38, 39, 36	ressply1bas2 22112	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
41		inss2 4201	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
42	40, 41	eqsstrdi 3991	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
43	42, 2	sseldd 3947	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
44	26	fveq1d 6860	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
45	2	fvresd 6878	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
46	44, 45	eqtr2d 2765	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
47	46	fveq1d 6860	. . . . 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 3947	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
50	26	fveq1d 6860	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
51	3	fvresd 6878	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
52	50, 51	eqtr2d 2765	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
53	52	fveq1d 6860	. . . . 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 22228	. . 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 1540 ∈ wcel 2109 Vcvv 3447 ∩ cin 3913 ↾ cres 5640 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 ↾_s cress 17200 +_gcplusg 17220 Ringcrg 20142 CRingccrg 20143 SubRingcsubrg 20478 PwSer₁cps1 22059 Poly₁cpl1 22061 evalSub₁ ces1 22200 eval₁ce1 22201

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-srg 20096 df-ring 20144 df-cring 20145 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-lmod 20768 df-lss 20838 df-lsp 20878 df-assa 21762 df-asp 21763 df-ascl 21764 df-psr 21818 df-mvr 21819 df-mpl 21820 df-opsr 21822 df-evls 21981 df-evl 21982 df-psr1 22064 df-vr1 22065 df-ply1 22066 df-coe1 22067 df-evls1 22202 df-evl1 22203

This theorem is referenced by: evls1maprhm 22263 cos9thpiminplylem6 33777

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