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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 19569 | . . . . . 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 | syl6eleq 2711 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅)) |
8 | 2 | ply1sca 19623 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊)) |
9 | 8 | eqcomd 2628 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (Scalar‘𝑊) = 𝑅) |
10 | 1, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (Scalar‘𝑊) = 𝑅) |
11 | 10 | fveq2d 6195 | . . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘𝑅)) |
12 | 7, 11 | eleqtrrd 2704 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝑊))) |
13 | crngring 18558 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
14 | 1, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | 2 | ply1ring 19618 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring) |
16 | 14, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring) |
17 | evl1varpw.g | . . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊) | |
18 | 17 | ringmgp 18553 | . . . . . . 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 2622 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
23 | 21, 2, 22 | vr1cl 19587 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑊)) |
24 | 14, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
25 | 17, 22 | mgpbas 18495 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝐺) |
26 | evl1varpw.e | . . . . . . 7 ⊢ ↑ = (.g‘𝐺) | |
27 | 25, 26 | mulgnn0cl 17558 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) |
28 | 19, 20, 24, 27 | syl3anc 1326 | . . . . 5 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) |
29 | eqid 2622 | . . . . . 6 ⊢ (algSc‘𝑊) = (algSc‘𝑊) | |
30 | eqid 2622 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
31 | eqid 2622 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
32 | eqid 2622 | . . . . . 6 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
33 | evl1scvarpw.t1 | . . . . . 6 ⊢ × = ( ·𝑠 ‘𝑊) | |
34 | 29, 30, 31, 22, 32, 33 | asclmul1 19339 | . . . . 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 1326 | . . . 4 ⊢ (𝜑 → (((algSc‘𝑊)‘𝐴)(.r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋))) |
36 | 35 | eqcomd 2628 | . . 3 ⊢ (𝜑 → (𝐴 × (𝑁 ↑ 𝑋)) = (((algSc‘𝑊)‘𝐴)(.r‘𝑊)(𝑁 ↑ 𝑋))) |
37 | 36 | fveq2d 6195 | . 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 19696 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑆)) |
41 | 1, 40 | syl 17 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑆)) |
42 | 2 | ply1lmod 19622 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod) |
43 | 14, 42 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
44 | 29, 30, 16, 43, 31, 22 | asclf 19337 | . . . 4 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶(Base‘𝑊)) |
45 | 44, 12 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → ((algSc‘𝑊)‘𝐴) ∈ (Base‘𝑊)) |
46 | evl1scvarpw.t2 | . . . 4 ⊢ ∙ = (.r‘𝑆) | |
47 | 22, 32, 46 | rhmmul 18727 | . . 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 1326 | . 2 ⊢ (𝜑 → (𝑄‘(((algSc‘𝑊)‘𝐴)(.r‘𝑊)(𝑁 ↑ 𝑋))) = ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋)))) |
49 | 38, 2, 6, 29 | evl1sca 19698 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐵) → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴})) |
50 | 1, 5, 49 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝑄‘((algSc‘𝑊)‘𝐴)) = (𝐵 × {𝐴})) |
51 | 38, 2, 17, 21, 6, 26, 1, 20 | evl1varpw 19725 | . . . 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 6194 | . . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘(𝑅 ↑s 𝐵)) |
55 | 53, 54 | eqtri 2644 | . . . . . . . . 9 ⊢ 𝑀 = (mulGrp‘(𝑅 ↑s 𝐵)) |
56 | 55 | fveq2i 6194 | . . . . . . . 8 ⊢ (.g‘𝑀) = (.g‘(mulGrp‘(𝑅 ↑s 𝐵))) |
57 | 52, 56 | eqtri 2644 | . . . . . . 7 ⊢ 𝐹 = (.g‘(mulGrp‘(𝑅 ↑s 𝐵))) |
58 | 57 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (.g‘(mulGrp‘(𝑅 ↑s 𝐵)))) |
59 | 58 | eqcomd 2628 | . . . . 5 ⊢ (𝜑 → (.g‘(mulGrp‘(𝑅 ↑s 𝐵))) = 𝐹) |
60 | 59 | oveqd 6667 | . . . 4 ⊢ (𝜑 → (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))(𝑄‘𝑋)) = (𝑁𝐹(𝑄‘𝑋))) |
61 | 51, 60 | eqtrd 2656 | . . 3 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁𝐹(𝑄‘𝑋))) |
62 | 50, 61 | oveq12d 6668 | . 2 ⊢ (𝜑 → ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋)))) |
63 | 37, 48, 62 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {csn 4177 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ℕ0cn0 11292 Basecbs 15857 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 ↑s cpws 16107 Mndcmnd 17294 .gcmg 17540 mulGrpcmgp 18489 Ringcrg 18547 CRingccrg 18548 RingHom crh 18712 LModclmod 18863 AssAlgcasa 19309 algSccascl 19311 var1cv1 19546 Poly1cpl1 19547 eval1ce1 19679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-evls 19506 df-evl 19507 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-evls1 19680 df-evl1 19681 |
This theorem is referenced by: (None) |
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