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Theorem evl1vsd 21420

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 2738	. . . 4 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
8		evl1vsd.4	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
9	1, 2, 3, 7, 4, 5, 8, 6	evl1scad 21411	. . 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 2738	. . 3 ⊢ (._r‘𝑃) = (._r‘𝑃)
12		evl1vsd.t	. . 3 ⊢ · = (._r‘𝑅)
13	1, 2, 3, 4, 5, 6, 9, 10, 11, 12	evl1muld 21419	. 2 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ∧ ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉)))
14	2	ply1assa 21280	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
15	5, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
16	2	ply1sca 21334	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
17	5, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
18	17	fveq2d 6760	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
19	3, 18	eqtrid 2790	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑃)))
20	8, 19	eleqtrd 2841	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Scalar‘𝑃)))
21	10	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
22		eqid 2738	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
23		eqid 2738	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
24		evl1vsd.s	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝑃)
25	7, 22, 23, 4, 11, 24	asclmul1 21000	. . . . 5 ⊢ ((𝑃 ∈ AssAlg ∧ 𝑁 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑀 ∈ 𝑈) → (((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) = (𝑁 ∙ 𝑀))
26	15, 20, 21, 25	syl3anc 1369	. . . 4 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) = (𝑁 ∙ 𝑀))
27	26	eleq1d 2823	. . 3 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ↔ (𝑁 ∙ 𝑀) ∈ 𝑈))
28	26	fveq2d 6760	. . . . 5 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀)) = (𝑂‘(𝑁 ∙ 𝑀)))
29	28	fveq1d 6758	. . . 4 ⊢ (𝜑 → ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌))
30	29	eqeq1d 2740	. . 3 ⊢ (𝜑 → (((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉) ↔ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))
31	27, 30	anbi12d 630	. 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 395 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 ._rcmulr 16889 Scalarcsca 16891 ·_𝑠 cvsca 16892 CRingccrg 19699 AssAlgcasa 20967 algSccascl 20969 Poly₁cpl1 21258 eval₁ce1 21390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-srg 19657 df-ring 19700 df-cring 19701 df-rnghom 19874 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-assa 20970 df-asp 20971 df-ascl 20972 df-psr 21022 df-mvr 21023 df-mpl 21024 df-opsr 21026 df-evls 21192 df-evl 21193 df-psr1 21261 df-ply1 21263 df-evl1 21392

This theorem is referenced by: evl1scvarpwval 21440 fta1blem 25238 plypf1 25278

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