evl1scvarpw - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evl1scvarpw		Structured version Visualization version GIF version

Theorem evl1scvarpw 21251

Description: Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)

**Hypotheses**
Ref	Expression
evl1varpw.q	⊢ 𝑄 = (eval₁‘𝑅)
evl1varpw.w	⊢ 𝑊 = (Poly₁‘𝑅)
evl1varpw.g	⊢ 𝐺 = (mulGrp‘𝑊)
evl1varpw.x	⊢ 𝑋 = (var₁‘𝑅)
evl1varpw.b	⊢ 𝐵 = (Base‘𝑅)
evl1varpw.e	⊢ ↑ = (._g‘𝐺)
evl1varpw.r	⊢ (𝜑 → 𝑅 ∈ CRing)
evl1varpw.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
evl1scvarpw.t1	⊢ × = ( ·_𝑠 ‘𝑊)
evl1scvarpw.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
evl1scvarpw.s	⊢ 𝑆 = (𝑅 ↑_s 𝐵)
evl1scvarpw.t2	⊢ ∙ = (._r‘𝑆)
evl1scvarpw.m	⊢ 𝑀 = (mulGrp‘𝑆)
evl1scvarpw.f	⊢ 𝐹 = (._g‘𝑀)

**Assertion**
Ref	Expression
evl1scvarpw	⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))

**Proof of Theorem evl1scvarpw**
Step	Hyp	Ref	Expression
1		evl1varpw.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
2		evl1varpw.w	. . . . . . 7 ⊢ 𝑊 = (Poly₁‘𝑅)
3	2	ply1assa 21092	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑊 ∈ AssAlg)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ AssAlg)
5		evl1scvarpw.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
6		evl1varpw.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
7	5, 6	eleqtrdi 2844	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
8	2	ply1sca 21146	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊))
9	8	eqcomd 2740	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (Scalar‘𝑊) = 𝑅)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (Scalar‘𝑊) = 𝑅)
11	10	fveq2d 6710	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘𝑅))
12	7, 11	eleqtrrd 2837	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝑊)))
13		crngring 19546	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
15	2	ply1ring 21141	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
17		evl1varpw.g	. . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊)
18	17	ringmgp 19540	. . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝐺 ∈ Mnd)
19	16, 18	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
20		evl1varpw.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21		evl1varpw.x	. . . . . . . 8 ⊢ 𝑋 = (var₁‘𝑅)
22		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
23	21, 2, 22	vr1cl 21110	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑊))
24	14, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
25	17, 22	mgpbas 19482	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝐺)
26		evl1varpw.e	. . . . . . 7 ⊢ ↑ = (._g‘𝐺)
27	25, 26	mulgnn0cl 18480	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
28	19, 20, 24, 27	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
29		eqid 2734	. . . . . 6 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
30		eqid 2734	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
31		eqid 2734	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
32		eqid 2734	. . . . . 6 ⊢ (._r‘𝑊) = (._r‘𝑊)
33		evl1scvarpw.t1	. . . . . 6 ⊢ × = ( ·_𝑠 ‘𝑊)
34	29, 30, 31, 22, 32, 33	asclmul1 20817	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐴 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) → (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋)))
35	4, 12, 28, 34	syl3anc 1373	. . . 4 ⊢ (𝜑 → (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋)))
36	35	eqcomd 2740	. . 3 ⊢ (𝜑 → (𝐴 × (𝑁 ↑ 𝑋)) = (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)))
37	36	fveq2d 6710	. 2 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = (𝑄‘(((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋))))
38		evl1varpw.q	. . . . 5 ⊢ 𝑄 = (eval₁‘𝑅)
39		evl1scvarpw.s	. . . . 5 ⊢ 𝑆 = (𝑅 ↑_s 𝐵)
40	38, 2, 39, 6	evl1rhm 21220	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑆))
41	1, 40	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑆))
42	2	ply1lmod 21145	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod)
43	14, 42	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
44	29, 30, 16, 43, 31, 22	asclf 20813	. . . 4 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
45	44, 12	ffvelrnd 6894	. . 3 ⊢ (𝜑 → ((algSc‘𝑊)‘𝐴) ∈ (Base‘𝑊))
46		evl1scvarpw.t2	. . . 4 ⊢ ∙ = (._r‘𝑆)
47	22, 32, 46	rhmmul 19719	. . 3 ⊢ ((𝑄 ∈ (𝑊 RingHom 𝑆) ∧ ((algSc‘𝑊)‘𝐴) ∈ (Base‘𝑊) ∧ (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) → (𝑄‘(((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋))) = ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))))
48	41, 45, 28, 47	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑄‘(((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋))) = ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))))
49	38, 2, 6, 29	evl1sca 21222	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐵) → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
50	1, 5, 49	syl2anc 587	. . 3 ⊢ (𝜑 → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
51	38, 2, 17, 21, 6, 26, 1, 20	evl1varpw 21249	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))(𝑄‘𝑋)))
52		evl1scvarpw.f	. . . . . . . 8 ⊢ 𝐹 = (._g‘𝑀)
53		evl1scvarpw.m	. . . . . . . . . 10 ⊢ 𝑀 = (mulGrp‘𝑆)
54	39	fveq2i 6709	. . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘(𝑅 ↑_s 𝐵))
55	53, 54	eqtri 2762	. . . . . . . . 9 ⊢ 𝑀 = (mulGrp‘(𝑅 ↑_s 𝐵))
56	55	fveq2i 6709	. . . . . . . 8 ⊢ (._g‘𝑀) = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
57	52, 56	eqtri 2762	. . . . . . 7 ⊢ 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))))
59	58	eqcomd 2740	. . . . 5 ⊢ (𝜑 → (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))) = 𝐹)
60	59	oveqd 7219	. . . 4 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))(𝑄‘𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
61	51, 60	eqtrd 2774	. . 3 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
62	50, 61	oveq12d 7220	. 2 ⊢ (𝜑 → ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))
63	37, 48, 62	3eqtrd 2778	1 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {csn 4531 × cxp 5538 ‘cfv 6369 (class class class)co 7202 ℕ₀cn0 12073 Basecbs 16684 ._rcmulr 16768 Scalarcsca 16770 ·_𝑠 cvsca 16771 ↑_s cpws 16923 Mndcmnd 18145 ._gcmg 18460 mulGrpcmgp 19476 Ringcrg 19534 CRingccrg 19535 RingHom crh 19704 LModclmod 19871 AssAlgcasa 20784 algSccascl 20786 var₁cv1 21069 Poly₁cpl1 21070 eval₁ce1 21202

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-ofr 7459 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-hom 16791 df-cco 16792 df-0g 16918 df-gsum 16919 df-prds 16924 df-pws 16926 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-srg 19493 df-ring 19536 df-cring 19537 df-rnghom 19707 df-subrg 19770 df-lmod 19873 df-lss 19941 df-lsp 19981 df-assa 20787 df-asp 20788 df-ascl 20789 df-psr 20840 df-mvr 20841 df-mpl 20842 df-opsr 20844 df-evls 21004 df-evl 21005 df-psr1 21073 df-vr1 21074 df-ply1 21075 df-evls1 21203 df-evl1 21204

This theorem is referenced by: (None)

W3C validator