evl1scvarpw - Metamath Proof Explorer

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Theorem evl1scvarpw 20509

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 20350	. . . . . 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 2923	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
8	2	ply1sca 20404	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊))
9	8	eqcomd 2827	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (Scalar‘𝑊) = 𝑅)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (Scalar‘𝑊) = 𝑅)
11	10	fveq2d 6660	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘𝑅))
12	7, 11	eleqtrrd 2916	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝑊)))
13		crngring 19291	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
15	2	ply1ring 20399	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
17		evl1varpw.g	. . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊)
18	17	ringmgp 19286	. . . . . . 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 2821	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
23	21, 2, 22	vr1cl 20368	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑊))
24	14, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
25	17, 22	mgpbas 19228	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝐺)
26		evl1varpw.e	. . . . . . 7 ⊢ ↑ = (._g‘𝐺)
27	25, 26	mulgnn0cl 18227	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
28	19, 20, 24, 27	syl3anc 1367	. . . . 5 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
29		eqid 2821	. . . . . 6 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
30		eqid 2821	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
31		eqid 2821	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
32		eqid 2821	. . . . . 6 ⊢ (._r‘𝑊) = (._r‘𝑊)
33		evl1scvarpw.t1	. . . . . 6 ⊢ × = ( ·_𝑠 ‘𝑊)
34	29, 30, 31, 22, 32, 33	asclmul1 20097	. . . . 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 1367	. . . 4 ⊢ (𝜑 → (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋)))
36	35	eqcomd 2827	. . 3 ⊢ (𝜑 → (𝐴 × (𝑁 ↑ 𝑋)) = (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)))
37	36	fveq2d 6660	. 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 20478	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑆))
41	1, 40	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑆))
42	2	ply1lmod 20403	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod)
43	14, 42	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
44	29, 30, 16, 43, 31, 22	asclf 20094	. . . 4 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
45	44, 12	ffvelrnd 6838	. . 3 ⊢ (𝜑 → ((algSc‘𝑊)‘𝐴) ∈ (Base‘𝑊))
46		evl1scvarpw.t2	. . . 4 ⊢ ∙ = (._r‘𝑆)
47	22, 32, 46	rhmmul 19462	. . 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 1367	. 2 ⊢ (𝜑 → (𝑄‘(((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋))) = ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))))
49	38, 2, 6, 29	evl1sca 20480	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐵) → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
50	1, 5, 49	syl2anc 586	. . 3 ⊢ (𝜑 → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
51	38, 2, 17, 21, 6, 26, 1, 20	evl1varpw 20507	. . . 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 6659	. . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘(𝑅 ↑_s 𝐵))
55	53, 54	eqtri 2844	. . . . . . . . 9 ⊢ 𝑀 = (mulGrp‘(𝑅 ↑_s 𝐵))
56	55	fveq2i 6659	. . . . . . . 8 ⊢ (._g‘𝑀) = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
57	52, 56	eqtri 2844	. . . . . . 7 ⊢ 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))))
59	58	eqcomd 2827	. . . . 5 ⊢ (𝜑 → (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))) = 𝐹)
60	59	oveqd 7159	. . . 4 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))(𝑄‘𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
61	51, 60	eqtrd 2856	. . 3 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
62	50, 61	oveq12d 7160	. 2 ⊢ (𝜑 → ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))
63	37, 48, 62	3eqtrd 2860	1 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {csn 4553 × cxp 5539 ‘cfv 6341 (class class class)co 7142 ℕ₀cn0 11884 Basecbs 16466 ._rcmulr 16549 Scalarcsca 16551 ·_𝑠 cvsca 16552 ↑_s cpws 16703 Mndcmnd 17894 ._gcmg 18207 mulGrpcmgp 19222 Ringcrg 19280 CRingccrg 19281 RingHom crh 19447 LModclmod 19617 AssAlgcasa 20065 algSccascl 20067 var₁cv1 20327 Poly₁cpl1 20328 eval₁ce1 20460

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-ofr 7396 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-sup 8892 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-fz 12883 df-fzo 13024 df-seq 13360 df-hash 13681 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-hom 16572 df-cco 16573 df-0g 16698 df-gsum 16699 df-prds 16704 df-pws 16706 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-mhm 17939 df-submnd 17940 df-grp 18089 df-minusg 18090 df-sbg 18091 df-mulg 18208 df-subg 18259 df-ghm 18339 df-cntz 18430 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-srg 19239 df-ring 19282 df-cring 19283 df-rnghom 19450 df-subrg 19516 df-lmod 19619 df-lss 19687 df-lsp 19727 df-assa 20068 df-asp 20069 df-ascl 20070 df-psr 20119 df-mvr 20120 df-mpl 20121 df-opsr 20123 df-evls 20269 df-evl 20270 df-psr1 20331 df-vr1 20332 df-ply1 20333 df-evls1 20461 df-evl1 20462

This theorem is referenced by: (None)

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