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Theorem evls1addd 22396

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 2740	. . . . . . 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 2740	. . . . . . 7 ⊢ ((Poly₁‘𝑆) ↾_s 𝐵) = ((Poly₁‘𝑆) ↾_s 𝐵)
10	4, 5, 6, 7, 8, 9	ressply1add 22252	. . . . . 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 7461	. . . . 5 ⊢ (𝑀 ⨣ 𝑁) = (𝑀(+_g‘𝑊)𝑁)
14	7	fvexi 6934	. . . . . . 7 ⊢ 𝐵 ∈ V
15		eqid 2740	. . . . . . . 8 ⊢ (+_g‘(Poly₁‘𝑆)) = (+_g‘(Poly₁‘𝑆))
16	9, 15	ressplusg 17349	. . . . . . 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 7461	. . . . 5 ⊢ (𝑀(+_g‘(Poly₁‘𝑆))𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁)
19	11, 13, 18	3eqtr4g 2805	. . . 4 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) = (𝑀(+_g‘(Poly₁‘𝑆))𝑁))
20	19	fveq2d 6924	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁)))
21	20	fveq1d 6922	. 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 2740	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
25		evls1addd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
26	22, 23, 6, 5, 7, 24, 25, 8	ressply1evl 22395	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
27	26	fveq1d 6922	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ⨣ 𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)))
28	5	subrgring 20602	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
29	6	ply1ring 22270	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
30	8, 28, 29	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
31	30	ringgrpd 20269	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
32	7, 12, 31, 2, 3	grpcld 18987	. . . . 5 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) ∈ 𝐵)
33	32	fvresd 6940	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)))
34	27, 33	eqtr2d 2781	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = (𝑄‘(𝑀 ⨣ 𝑁)))
35	34	fveq1d 6922	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁))‘𝐶) = ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶))
36		eqid 2740	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
37		evls1addd.y	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
38		eqid 2740	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
39		eqid 2740	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
40	4, 5, 6, 7, 8, 38, 39, 36	ressply1bas2 22250	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
41		inss2 4259	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
42	40, 41	eqsstrdi 4063	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
43	42, 2	sseldd 4009	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
44	26	fveq1d 6922	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
45	2	fvresd 6940	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
46	44, 45	eqtr2d 2781	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
47	46	fveq1d 6922	. . . . 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 4009	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
50	26	fveq1d 6922	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
51	3	fvresd 6940	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
52	50, 51	eqtr2d 2781	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
53	52	fveq1d 6922	. . . . 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 22366	. . 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 2788	1 ⊢ (𝜑 → ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) + ((𝑄‘𝑁)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ∩ cin 3975 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 ↾_s cress 17287 +_gcplusg 17311 Ringcrg 20260 CRingccrg 20261 SubRingcsubrg 20595 PwSer₁cps1 22197 Poly₁cpl1 22199 evalSub₁ ces1 22338 eval₁ce1 22339

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-lmod 20882 df-lss 20953 df-lsp 20993 df-assa 21896 df-asp 21897 df-ascl 21898 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-evls 22121 df-evl 22122 df-psr1 22202 df-vr1 22203 df-ply1 22204 df-coe1 22205 df-evls1 22340 df-evl1 22341

This theorem is referenced by: evls1maprhm 22401

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