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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 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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