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Theorem evls1addd 22391

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 2735	. . . . . . 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 2735	. . . . . . 7 ⊢ ((Poly₁‘𝑆) ↾_s 𝐵) = ((Poly₁‘𝑆) ↾_s 𝐵)
10	4, 5, 6, 7, 8, 9	ressply1add 22247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵)) → (𝑀(+_g‘𝑊)𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
11	1, 2, 3, 10	syl12anc 837	. . . . 5 ⊢ (𝜑 → (𝑀(+_g‘𝑊)𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
12		evls1addd.1	. . . . . 6 ⊢ ⨣ = (+_g‘𝑊)
13	12	oveqi 7444	. . . . 5 ⊢ (𝑀 ⨣ 𝑁) = (𝑀(+_g‘𝑊)𝑁)
14	7	fvexi 6921	. . . . . . 7 ⊢ 𝐵 ∈ V
15		eqid 2735	. . . . . . . 8 ⊢ (+_g‘(Poly₁‘𝑆)) = (+_g‘(Poly₁‘𝑆))
16	9, 15	ressplusg 17336	. . . . . . 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 7444	. . . . 5 ⊢ (𝑀(+_g‘(Poly₁‘𝑆))𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁)
19	11, 13, 18	3eqtr4g 2800	. . . 4 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) = (𝑀(+_g‘(Poly₁‘𝑆))𝑁))
20	19	fveq2d 6911	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁)))
21	20	fveq1d 6909	. 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 2735	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
25		evls1addd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
26	22, 23, 6, 5, 7, 24, 25, 8	ressply1evl 22390	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
27	26	fveq1d 6909	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ⨣ 𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)))
28	5	subrgring 20591	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
29	6	ply1ring 22265	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
30	8, 28, 29	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
31	30	ringgrpd 20260	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
32	7, 12, 31, 2, 3	grpcld 18978	. . . . 5 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) ∈ 𝐵)
33	32	fvresd 6927	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)))
34	27, 33	eqtr2d 2776	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = (𝑄‘(𝑀 ⨣ 𝑁)))
35	34	fveq1d 6909	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁))‘𝐶) = ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶))
36		eqid 2735	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
37		evls1addd.y	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
38		eqid 2735	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
39		eqid 2735	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
40	4, 5, 6, 7, 8, 38, 39, 36	ressply1bas2 22245	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
41		inss2 4246	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
42	40, 41	eqsstrdi 4050	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
43	42, 2	sseldd 3996	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
44	26	fveq1d 6909	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
45	2	fvresd 6927	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
46	44, 45	eqtr2d 2776	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
47	46	fveq1d 6909	. . . . 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 3996	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
50	26	fveq1d 6909	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
51	3	fvresd 6927	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
52	50, 51	eqtr2d 2776	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
53	52	fveq1d 6909	. . . . 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 22361	. . 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 2783	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 2106 Vcvv 3478 ∩ cin 3962 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾_s cress 17274 +_gcplusg 17298 Ringcrg 20251 CRingccrg 20252 SubRingcsubrg 20586 PwSer₁cps1 22192 Poly₁cpl1 22194 evalSub₁ ces1 22333 eval₁ce1 22334

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-evl1 22336

This theorem is referenced by: evls1maprhm 22396

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