evls1addd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evls1addd		Structured version Visualization version GIF version

Theorem evls1addd 22319

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 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		evls1addd.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	ressply1add 22174	. . . . . 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 7373	. . . . 5 ⊢ (𝑀 ⨣ 𝑁) = (𝑀(+_g‘𝑊)𝑁)
14	7	fvexi 6849	. . . . . . 7 ⊢ 𝐵 ∈ V
15		eqid 2737	. . . . . . . 8 ⊢ (+_g‘(Poly₁‘𝑆)) = (+_g‘(Poly₁‘𝑆))
16	9, 15	ressplusg 17215	. . . . . . 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 7373	. . . . 5 ⊢ (𝑀(+_g‘(Poly₁‘𝑆))𝑁) = (𝑀(+_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁)
19	11, 13, 18	3eqtr4g 2797	. . . 4 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) = (𝑀(+_g‘(Poly₁‘𝑆))𝑁))
20	19	fveq2d 6839	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀(+_g‘(Poly₁‘𝑆))𝑁)))
21	20	fveq1d 6837	. 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 2737	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
25		evls1addd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
26	22, 23, 6, 5, 7, 24, 25, 8	ressply1evl 22318	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
27	26	fveq1d 6837	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ⨣ 𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)))
28	5	subrgring 20511	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
29	6	ply1ring 22192	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
30	8, 28, 29	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
31	30	ringgrpd 20181	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
32	7, 12, 31, 2, 3	grpcld 18881	. . . . 5 ⊢ (𝜑 → (𝑀 ⨣ 𝑁) ∈ 𝐵)
33	32	fvresd 6855	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀 ⨣ 𝑁)) = ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)))
34	27, 33	eqtr2d 2773	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁)) = (𝑄‘(𝑀 ⨣ 𝑁)))
35	34	fveq1d 6837	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀 ⨣ 𝑁))‘𝐶) = ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶))
36		eqid 2737	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
37		evls1addd.y	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
38		eqid 2737	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
39		eqid 2737	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
40	4, 5, 6, 7, 8, 38, 39, 36	ressply1bas2 22172	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
41		inss2 4191	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
42	40, 41	eqsstrdi 3979	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
43	42, 2	sseldd 3935	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
44	26	fveq1d 6837	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
45	2	fvresd 6855	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
46	44, 45	eqtr2d 2773	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
47	46	fveq1d 6837	. . . . 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 3935	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
50	26	fveq1d 6837	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
51	3	fvresd 6855	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
52	50, 51	eqtr2d 2773	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
53	52	fveq1d 6837	. . . . 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 22289	. . 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 2780	1 ⊢ (𝜑 → ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) + ((𝑄‘𝑁)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3441 ∩ cin 3901 ↾ cres 5627 ‘cfv 6493 (class class class)co 7360 Basecbs 17140 ↾_s cress 17161 +_gcplusg 17181 Ringcrg 20172 CRingccrg 20173 SubRingcsubrg 20506 PwSer₁cps1 22119 Poly₁cpl1 22121 evalSub₁ ces1 22261 eval₁ce1 22262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-fz 13428 df-fzo 13575 df-seq 13929 df-hash 14258 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-hom 17205 df-cco 17206 df-0g 17365 df-gsum 17366 df-prds 17371 df-pws 17373 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-cntz 19250 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-cring 20175 df-rhm 20412 df-subrng 20483 df-subrg 20507 df-lmod 20817 df-lss 20887 df-lsp 20927 df-assa 21812 df-asp 21813 df-ascl 21814 df-psr 21869 df-mvr 21870 df-mpl 21871 df-opsr 21873 df-evls 22033 df-evl 22034 df-psr1 22124 df-vr1 22125 df-ply1 22126 df-coe1 22127 df-evls1 22263 df-evl1 22264

This theorem is referenced by: evls1maprhm 22324 cos9thpiminplylem6 33946

W3C validator