evl1scvarpw - Metamath Proof Explorer

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Theorem evl1scvarpw 21557

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 21398	. . . . . 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 21452	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊))
9	8	eqcomd 2739	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (Scalar‘𝑊) = 𝑅)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (Scalar‘𝑊) = 𝑅)
11	10	fveq2d 6796	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘𝑅))
12	7, 11	eleqtrrd 2837	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝑊)))
13		crngring 19823	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
15	2	ply1ring 21447	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
17		evl1varpw.g	. . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊)
18	17	ringmgp 19817	. . . . . . 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 2733	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
23	21, 2, 22	vr1cl 21416	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑊))
24	14, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
25	17, 22	mgpbas 19754	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝐺)
26		evl1varpw.e	. . . . . . 7 ⊢ ↑ = (._g‘𝐺)
27	25, 26	mulgnn0cl 18748	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
28	19, 20, 24, 27	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
29		eqid 2733	. . . . . 6 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
30		eqid 2733	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
31		eqid 2733	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
32		eqid 2733	. . . . . 6 ⊢ (._r‘𝑊) = (._r‘𝑊)
33		evl1scvarpw.t1	. . . . . 6 ⊢ × = ( ·_𝑠 ‘𝑊)
34	29, 30, 31, 22, 32, 33	asclmul1 21118	. . . . 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 1369	. . . 4 ⊢ (𝜑 → (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋)))
36	35	eqcomd 2739	. . 3 ⊢ (𝜑 → (𝐴 × (𝑁 ↑ 𝑋)) = (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)))
37	36	fveq2d 6796	. 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 21526	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑆))
41	1, 40	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑆))
42	2	ply1lmod 21451	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod)
43	14, 42	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
44	29, 30, 16, 43, 31, 22	asclf 21114	. . . 4 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
45	44, 12	ffvelcdmd 6982	. . 3 ⊢ (𝜑 → ((algSc‘𝑊)‘𝐴) ∈ (Base‘𝑊))
46		evl1scvarpw.t2	. . . 4 ⊢ ∙ = (._r‘𝑆)
47	22, 32, 46	rhmmul 19999	. . 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 1369	. 2 ⊢ (𝜑 → (𝑄‘(((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋))) = ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))))
49	38, 2, 6, 29	evl1sca 21528	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐵) → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
50	1, 5, 49	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
51	38, 2, 17, 21, 6, 26, 1, 20	evl1varpw 21555	. . . 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 6795	. . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘(𝑅 ↑_s 𝐵))
55	53, 54	eqtri 2761	. . . . . . . . 9 ⊢ 𝑀 = (mulGrp‘(𝑅 ↑_s 𝐵))
56	55	fveq2i 6795	. . . . . . . 8 ⊢ (._g‘𝑀) = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
57	52, 56	eqtri 2761	. . . . . . 7 ⊢ 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))))
59	58	eqcomd 2739	. . . . 5 ⊢ (𝜑 → (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))) = 𝐹)
60	59	oveqd 7312	. . . 4 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))(𝑄‘𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
61	51, 60	eqtrd 2773	. . 3 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
62	50, 61	oveq12d 7313	. 2 ⊢ (𝜑 → ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))
63	37, 48, 62	3eqtrd 2777	1 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 {csn 4564 × cxp 5589 ‘cfv 6447 (class class class)co 7295 ℕ₀cn0 12261 Basecbs 16940 ._rcmulr 16991 Scalarcsca 16993 ·_𝑠 cvsca 16994 ↑_s cpws 17185 Mndcmnd 18413 ._gcmg 18728 mulGrpcmgp 19748 Ringcrg 19811 CRingccrg 19812 RingHom crh 19984 LModclmod 20151 AssAlgcasa 21085 algSccascl 21087 var₁cv1 21375 Poly₁cpl1 21376 eval₁ce1 21508

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-ofr 7554 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-pm 8638 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-sup 9229 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-fzo 13411 df-seq 13750 df-hash 14073 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-sca 17006 df-vsca 17007 df-ip 17008 df-tset 17009 df-ple 17010 df-ds 17012 df-hom 17014 df-cco 17015 df-0g 17180 df-gsum 17181 df-prds 17186 df-pws 17188 df-mre 17323 df-mrc 17324 df-acs 17326 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-mhm 18458 df-submnd 18459 df-grp 18608 df-minusg 18609 df-sbg 18610 df-mulg 18729 df-subg 18780 df-ghm 18860 df-cntz 18951 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-srg 19770 df-ring 19813 df-cring 19814 df-rnghom 19987 df-subrg 20050 df-lmod 20153 df-lss 20222 df-lsp 20262 df-assa 21088 df-asp 21089 df-ascl 21090 df-psr 21140 df-mvr 21141 df-mpl 21142 df-opsr 21144 df-evls 21310 df-evl 21311 df-psr1 21379 df-vr1 21380 df-ply1 21381 df-evls1 21509 df-evl1 21510

This theorem is referenced by: (None)

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