evl1vsd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evl1vsd		Structured version Visualization version GIF version

Theorem evl1vsd 21710

Description: Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)

**Hypotheses**
Ref	Expression
evl1addd.q	⊢ 𝑂 = (eval₁‘𝑅)
evl1addd.p	⊢ 𝑃 = (Poly₁‘𝑅)
evl1addd.b	⊢ 𝐵 = (Base‘𝑅)
evl1addd.u	⊢ 𝑈 = (Base‘𝑃)
evl1addd.1	⊢ (𝜑 → 𝑅 ∈ CRing)
evl1addd.2	⊢ (𝜑 → 𝑌 ∈ 𝐵)
evl1addd.3	⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))
evl1vsd.4	⊢ (𝜑 → 𝑁 ∈ 𝐵)
evl1vsd.s	⊢ ∙ = ( ·_𝑠 ‘𝑃)
evl1vsd.t	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
evl1vsd	⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))

**Proof of Theorem evl1vsd**
Step	Hyp	Ref	Expression
1		evl1addd.q	. . 3 ⊢ 𝑂 = (eval₁‘𝑅)
2		evl1addd.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
3		evl1addd.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4		evl1addd.u	. . 3 ⊢ 𝑈 = (Base‘𝑃)
5		evl1addd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
6		evl1addd.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		eqid 2736	. . . 4 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
8		evl1vsd.4	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
9	1, 2, 3, 7, 4, 5, 8, 6	evl1scad 21701	. . 3 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑁) ∈ 𝑈 ∧ ((𝑂‘((algSc‘𝑃)‘𝑁))‘𝑌) = 𝑁))
10		evl1addd.3	. . 3 ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))
11		eqid 2736	. . 3 ⊢ (._r‘𝑃) = (._r‘𝑃)
12		evl1vsd.t	. . 3 ⊢ · = (._r‘𝑅)
13	1, 2, 3, 4, 5, 6, 9, 10, 11, 12	evl1muld 21709	. 2 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ∧ ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉)))
14	2	ply1assa 21570	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
15	5, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
16	2	ply1sca 21624	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
17	5, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
18	17	fveq2d 6846	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
19	3, 18	eqtrid 2788	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑃)))
20	8, 19	eleqtrd 2840	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Scalar‘𝑃)))
21	10	simpld 495	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
22		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
23		eqid 2736	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
24		evl1vsd.s	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝑃)
25	7, 22, 23, 4, 11, 24	asclmul1 21289	. . . . 5 ⊢ ((𝑃 ∈ AssAlg ∧ 𝑁 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑀 ∈ 𝑈) → (((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) = (𝑁 ∙ 𝑀))
26	15, 20, 21, 25	syl3anc 1371	. . . 4 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) = (𝑁 ∙ 𝑀))
27	26	eleq1d 2822	. . 3 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ↔ (𝑁 ∙ 𝑀) ∈ 𝑈))
28	26	fveq2d 6846	. . . . 5 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀)) = (𝑂‘(𝑁 ∙ 𝑀)))
29	28	fveq1d 6844	. . . 4 ⊢ (𝜑 → ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌))
30	29	eqeq1d 2738	. . 3 ⊢ (𝜑 → (((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉) ↔ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))
31	27, 30	anbi12d 631	. 2 ⊢ (𝜑 → (((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ∧ ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉)) ↔ ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉))))
32	13, 31	mpbid 231	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 ._rcmulr 17134 Scalarcsca 17136 ·_𝑠 cvsca 17137 CRingccrg 19965 AssAlgcasa 21256 algSccascl 21258 Poly₁cpl1 21548 eval₁ce1 21680

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-opsr 21315 df-evls 21482 df-evl 21483 df-psr1 21551 df-ply1 21553 df-evl1 21682

This theorem is referenced by: evl1scvarpwval 21730 fta1blem 25533 plypf1 25573

W3C validator