evl1scvarpw - Metamath Proof Explorer

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Theorem evl1scvarpw 21527

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 21368	. . . . . 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 2851	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
8	2	ply1sca 21422	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊))
9	8	eqcomd 2746	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (Scalar‘𝑊) = 𝑅)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (Scalar‘𝑊) = 𝑅)
11	10	fveq2d 6775	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘𝑅))
12	7, 11	eleqtrrd 2844	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝑊)))
13		crngring 19793	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
15	2	ply1ring 21417	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
17		evl1varpw.g	. . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊)
18	17	ringmgp 19787	. . . . . . 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 2740	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
23	21, 2, 22	vr1cl 21386	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑊))
24	14, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
25	17, 22	mgpbas 19724	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝐺)
26		evl1varpw.e	. . . . . . 7 ⊢ ↑ = (._g‘𝐺)
27	25, 26	mulgnn0cl 18718	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
28	19, 20, 24, 27	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊))
29		eqid 2740	. . . . . 6 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
30		eqid 2740	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
31		eqid 2740	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
32		eqid 2740	. . . . . 6 ⊢ (._r‘𝑊) = (._r‘𝑊)
33		evl1scvarpw.t1	. . . . . 6 ⊢ × = ( ·_𝑠 ‘𝑊)
34	29, 30, 31, 22, 32, 33	asclmul1 21088	. . . . 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 1370	. . . 4 ⊢ (𝜑 → (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋)))
36	35	eqcomd 2746	. . 3 ⊢ (𝜑 → (𝐴 × (𝑁 ↑ 𝑋)) = (((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋)))
37	36	fveq2d 6775	. 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 21496	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑆))
41	1, 40	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑆))
42	2	ply1lmod 21421	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod)
43	14, 42	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
44	29, 30, 16, 43, 31, 22	asclf 21084	. . . 4 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
45	44, 12	ffvelrnd 6959	. . 3 ⊢ (𝜑 → ((algSc‘𝑊)‘𝐴) ∈ (Base‘𝑊))
46		evl1scvarpw.t2	. . . 4 ⊢ ∙ = (._r‘𝑆)
47	22, 32, 46	rhmmul 19969	. . 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 1370	. 2 ⊢ (𝜑 → (𝑄‘(((algSc‘𝑊)‘𝐴)(._r‘𝑊)(𝑁 ↑ 𝑋))) = ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))))
49	38, 2, 6, 29	evl1sca 21498	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐵) → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
50	1, 5, 49	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴}))
51	38, 2, 17, 21, 6, 26, 1, 20	evl1varpw 21525	. . . 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 6774	. . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘(𝑅 ↑_s 𝐵))
55	53, 54	eqtri 2768	. . . . . . . . 9 ⊢ 𝑀 = (mulGrp‘(𝑅 ↑_s 𝐵))
56	55	fveq2i 6774	. . . . . . . 8 ⊢ (._g‘𝑀) = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
57	52, 56	eqtri 2768	. . . . . . 7 ⊢ 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐹 = (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))))
59	58	eqcomd 2746	. . . . 5 ⊢ (𝜑 → (._g‘(mulGrp‘(𝑅 ↑_s 𝐵))) = 𝐹)
60	59	oveqd 7288	. . . 4 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(𝑅 ↑_s 𝐵)))(𝑄‘𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
61	51, 60	eqtrd 2780	. . 3 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁𝐹(𝑄‘𝑋)))
62	50, 61	oveq12d 7289	. 2 ⊢ (𝜑 → ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))
63	37, 48, 62	3eqtrd 2784	1 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 {csn 4567 × cxp 5588 ‘cfv 6432 (class class class)co 7271 ℕ₀cn0 12233 Basecbs 16910 ._rcmulr 16961 Scalarcsca 16963 ·_𝑠 cvsca 16964 ↑_s cpws 17155 Mndcmnd 18383 ._gcmg 18698 mulGrpcmgp 19718 Ringcrg 19781 CRingccrg 19782 RingHom crh 19954 LModclmod 20121 AssAlgcasa 21055 algSccascl 21057 var₁cv1 21345 Poly₁cpl1 21346 eval₁ce1 21478

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 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-ofr 7528 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-fzo 13382 df-seq 13720 df-hash 14043 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-hom 16984 df-cco 16985 df-0g 17150 df-gsum 17151 df-prds 17156 df-pws 17158 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-srg 19740 df-ring 19783 df-cring 19784 df-rnghom 19957 df-subrg 20020 df-lmod 20123 df-lss 20192 df-lsp 20232 df-assa 21058 df-asp 21059 df-ascl 21060 df-psr 21110 df-mvr 21111 df-mpl 21112 df-opsr 21114 df-evls 21280 df-evl 21281 df-psr1 21349 df-vr1 21350 df-ply1 21351 df-evls1 21479 df-evl1 21480

This theorem is referenced by: (None)

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