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Mirrors > Home > MPE Home > Th. List > evl1scvarpw | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
evl1scvarpw | ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋)))) |
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) |
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