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Theorem evl1vsd 22394

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 2761	. . . 4 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
8		evl1vsd.4	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
9	1, 2, 3, 7, 4, 5, 8, 6	evl1scad 22385	. . 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 2761	. . 3 ⊢ (._r‘𝑃) = (._r‘𝑃)
12		evl1vsd.t	. . 3 ⊢ · = (._r‘𝑅)
13	1, 2, 3, 4, 5, 6, 9, 10, 11, 12	evl1muld 22393	. 2 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ∧ ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉)))
14	2	ply1assa 22248	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
15	5, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
16	2	ply1sca 22301	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
17	5, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
18	17	fveq2d 6865	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
19	3, 18	eqtrid 2808	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑃)))
20	8, 19	eleqtrd 2863	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Scalar‘𝑃)))
21	10	simpld 498	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
22		eqid 2761	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
23		eqid 2761	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
24		evl1vsd.s	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝑃)
25	7, 22, 23, 4, 11, 24	asclmul1 21925	. . . . 5 ⊢ ((𝑃 ∈ AssAlg ∧ 𝑁 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑀 ∈ 𝑈) → (((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) = (𝑁 ∙ 𝑀))
26	15, 20, 21, 25	syl3anc 1389	. . . 4 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) = (𝑁 ∙ 𝑀))
27	26	eleq1d 2846	. . 3 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀) ∈ 𝑈 ↔ (𝑁 ∙ 𝑀) ∈ 𝑈))
28	26	fveq2d 6865	. . . . 5 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀)) = (𝑂‘(𝑁 ∙ 𝑀)))
29	28	fveq1d 6863	. . . 4 ⊢ (𝜑 → ((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌))
30	29	eqeq1d 2763	. . 3 ⊢ (𝜑 → (((𝑂‘(((algSc‘𝑃)‘𝑁)(._r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉) ↔ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))
31	27, 30	anbi12d 641	. 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 234	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ‘cfv 6515 (class class class)co 7390 Basecbs 17235 ._rcmulr 17277 Scalarcsca 17279 ·_𝑠 cvsca 17280 CRingccrg 20270 AssAlgcasa 21889 algSccascl 21891 Poly₁cpl1 22226 eval₁ce1 22364

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-ofr 7655 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-sup 9381 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-fz 13506 df-fzo 13653 df-seq 14008 df-hash 14337 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-grp 18968 df-minusg 18969 df-sbg 18970 df-mulg 19100 df-subg 19155 df-ghm 19244 df-cntz 19347 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-srg 20223 df-ring 20271 df-cring 20272 df-rhm 20507 df-subrng 20582 df-subrg 20606 df-lmod 20916 df-lss 20986 df-lsp 21026 df-assa 21892 df-asp 21893 df-ascl 21894 df-psr 21948 df-mvr 21949 df-mpl 21950 df-opsr 21952 df-evls 22114 df-evl 22115 df-psr1 22229 df-ply1 22231 df-evl1 22366

This theorem is referenced by: evl1scvarpwval 22414 evls1vsca 22423 fta1blem 26218 plypf1 26259

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