evl1vsd - Metamath Proof Explorer

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Theorem evl1vsd 22260

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 2731	. . . 4 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
8		evl1vsd.4	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
9	1, 2, 3, 7, 4, 5, 8, 6	evl1scad 22251	. . 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 2731	. . 3 ⊢ (._r‘𝑃) = (._r‘𝑃)
12		evl1vsd.t	. . 3 ⊢ · = (._r‘𝑅)
13	1, 2, 3, 4, 5, 6, 9, 10, 11, 12	evl1muld 22259	. 2 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ∧ ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉)))
14	2	ply1assa 22113	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
15	5, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
16	2	ply1sca 22166	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
17	5, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
18	17	fveq2d 6826	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
19	3, 18	eqtrid 2778	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑃)))
20	8, 19	eleqtrd 2833	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Scalar‘𝑃)))
21	10	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
22		eqid 2731	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
23		eqid 2731	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
24		evl1vsd.s	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝑃)
25	7, 22, 23, 4, 11, 24	asclmul1 21824	. . . . 5 ⊢ ((𝑃 ∈ AssAlg ∧ 𝑁 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑀 ∈ 𝑈) → (((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) = (𝑁 ∙ 𝑀))
26	15, 20, 21, 25	syl3anc 1373	. . . 4 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) = (𝑁 ∙ 𝑀))
27	26	eleq1d 2816	. . 3 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ↔ (𝑁 ∙ 𝑀) ∈ 𝑈))
28	26	fveq2d 6826	. . . . 5 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀)) = (𝑂‘(𝑁 ∙ 𝑀)))
29	28	fveq1d 6824	. . . 4 ⊢ (𝜑 → ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌))
30	29	eqeq1d 2733	. . 3 ⊢ (𝜑 → (((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉) ↔ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))
31	27, 30	anbi12d 632	. 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 232	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 ._rcmulr 17162 Scalarcsca 17164 ·_𝑠 cvsca 17165 CRingccrg 20153 AssAlgcasa 21788 algSccascl 21790 Poly₁cpl1 22090 eval₁ce1 22230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19126 df-cntz 19230 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-srg 20106 df-ring 20154 df-cring 20155 df-rhm 20391 df-subrng 20462 df-subrg 20486 df-lmod 20796 df-lss 20866 df-lsp 20906 df-assa 21791 df-asp 21792 df-ascl 21793 df-psr 21847 df-mvr 21848 df-mpl 21849 df-opsr 21851 df-evls 22010 df-evl 22011 df-psr1 22093 df-ply1 22095 df-evl1 22232

This theorem is referenced by: evl1scvarpwval 22280 evls1vsca 22289 fta1blem 26104 plypf1 26145

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