evl1scvarpw - Metamath Proof Explorer

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Theorem evl1scvarpw 22349

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 22184	. . . . . 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 2849	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
8	2	ply1sca 22237	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊))
9	8	eqcomd 2745	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (Scalar‘𝑊) = 𝑅)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (Scalar‘𝑊) = 𝑅)
11	10	fveq2d 6831	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘𝑅))
12	7, 11	eleqtrrd 2842	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝑊)))
13		evl1varpw.g	. . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑊)
14		eqid 2739	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
15	13, 14	mgpbas 20117	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝐺)
16		evl1varpw.e	. . . . . 6 ⊢ ↑ = (._g‘𝐺)
17		crngring 20217	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18	1, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
19	2	ply1ring 22232	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring)
20	18, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
21	13	ringmgp 20211	. . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝐺 ∈ Mnd)
22	20, 21	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
23		evl1varpw.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24		evl1varpw.x	. . . . . . . 8 ⊢ 𝑋 = (var₁‘𝑅)
25	24, 2, 14	vr1cl 22202	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑊))
26	18, 25	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
27	15, 16, 22, 23, 26	mulgnn0cld 19062	. . . . 5 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
28		eqid 2739	. . . . . 6 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
29		eqid 2739	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
30		eqid 2739	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31		eqid 2739	. . . . . 6 ⊢ (._r‘𝑊) = (._r‘𝑊)
32		evl1scvarpw.t1	. . . . . 6 ⊢ × = ( ·_𝑠 ‘𝑊)
33	28, 29, 30, 14, 31, 32	asclmul1 21861	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐴 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) → (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋)))
34	4, 12, 27, 33	syl3anc 1379	. . . 4 ⊢ (𝜑 → (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋)))
35	34	eqcomd 2745	. . 3 ⊢ (𝜑 → (𝐴 × (𝑁 ↑ 𝑋)) = (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)))
36	35	fveq2d 6831	. 2 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = (𝑄‘(((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋))))
37		evl1varpw.q	. . . . 5 ⊢ 𝑄 = (eval₁‘𝑅)
38		evl1scvarpw.s	. . . . 5 ⊢ 𝑆 = (𝑅 ↑_s 𝐵)
39	37, 2, 38, 6	evl1rhm 22318	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑆))
40	1, 39	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑆))
41	2	ply1lmod 22236	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod)
42	18, 41	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
43	28, 29, 20, 42, 30, 14	asclf 21856	. . . 4 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
44	43, 12	ffvelcdmd 7026	. . 3 ⊢ (𝜑 → ((algSc‘𝑊)‘𝐴) ∈ (Base‘𝑊))
45		evl1scvarpw.t2	. . . 4 ⊢ ∙ = (._r‘𝑆)
46	14, 31, 45	rhmmul 20457	. . 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))))
47	40, 44, 27, 46	syl3anc 1379	. 2 ⊢ (𝜑 → (𝑄‘(((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋))) = ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))))
48	37, 2, 6, 28	evl1sca 22320	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐵) → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
49	1, 5, 48	syl2anc 590	. . 3 ⊢ (𝜑 → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
50	37, 2, 13, 24, 6, 16, 1, 23	evl1varpw 22347	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))(𝑄‘𝑋)))
51		evl1scvarpw.f	. . . . . . . 8 ⊢ 𝐹 = (._g‘𝑀)
52		evl1scvarpw.m	. . . . . . . . . 10 ⊢ 𝑀 = (mulGrp‘𝑆)
53	38	fveq2i 6830	. . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘(𝑅 ↑_s 𝐵))
54	52, 53	eqtri 2762	. . . . . . . . 9 ⊢ 𝑀 = (mulGrp‘(𝑅 ↑_s 𝐵))
55	54	fveq2i 6830	. . . . . . . 8 ⊢ (._g‘𝑀) = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
56	51, 55	eqtri 2762	. . . . . . 7 ⊢ 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
57	56	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))))
58	57	eqcomd 2745	. . . . 5 ⊢ (𝜑 → (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))) = 𝐹)
59	58	oveqd 7373	. . . 4 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))(𝑄‘𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
60	50, 59	eqtrd 2774	. . 3 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
61	49, 60	oveq12d 7374	. 2 ⊢ (𝜑 → ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))
62	36, 47, 61	3eqtrd 2778	1 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 {csn 4555 × cxp 5616 ‘cfv 6485 (class class class)co 7356 ℕ₀cn0 12428 Basecbs 17170 ._rcmulr 17212 Scalarcsca 17214 ·_𝑠 cvsca 17215 ↑_s cpws 17400 Mndcmnd 18693 ._gcmg 19034 mulGrpcmgp 20112 Ringcrg 20205 CRingccrg 20206 RingHom crh 20440 LModclmod 20850 AssAlgcasa 21825 algSccascl 21827 var₁cv1 22161 Poly₁cpl1 22162 eval₁ce1 22300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-srg 20159 df-ring 20207 df-cring 20208 df-rhm 20443 df-subrng 20518 df-subrg 20542 df-lmod 20852 df-lss 20922 df-lsp 20962 df-assa 21828 df-asp 21829 df-ascl 21830 df-psr 21884 df-mvr 21885 df-mpl 21886 df-opsr 21888 df-evls 22050 df-evl 22051 df-psr1 22165 df-vr1 22166 df-ply1 22167 df-evls1 22301 df-evl1 22302

This theorem is referenced by: (None)

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