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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 20295 | . . . . . 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 2920 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅)) |
8 | 2 | ply1sca 20349 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊)) |
9 | 8 | eqcomd 2824 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (Scalar‘𝑊) = 𝑅) |
10 | 1, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (Scalar‘𝑊) = 𝑅) |
11 | 10 | fveq2d 6667 | . . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘𝑅)) |
12 | 7, 11 | eleqtrrd 2913 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝑊))) |
13 | crngring 19237 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
14 | 1, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | 2 | ply1ring 20344 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring) |
16 | 14, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring) |
17 | evl1varpw.g | . . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊) | |
18 | 17 | ringmgp 19232 | . . . . . . 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 2818 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
23 | 21, 2, 22 | vr1cl 20313 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑊)) |
24 | 14, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
25 | 17, 22 | mgpbas 19174 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝐺) |
26 | evl1varpw.e | . . . . . . 7 ⊢ ↑ = (.g‘𝐺) | |
27 | 25, 26 | mulgnn0cl 18182 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) |
28 | 19, 20, 24, 27 | syl3anc 1363 | . . . . 5 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) |
29 | eqid 2818 | . . . . . 6 ⊢ (algSc‘𝑊) = (algSc‘𝑊) | |
30 | eqid 2818 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
31 | eqid 2818 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
32 | eqid 2818 | . . . . . 6 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
33 | evl1scvarpw.t1 | . . . . . 6 ⊢ × = ( ·𝑠 ‘𝑊) | |
34 | 29, 30, 31, 22, 32, 33 | asclmul1 20042 | . . . . 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 1363 | . . . 4 ⊢ (𝜑 → (((algSc‘𝑊)‘𝐴)(.r‘𝑊)(𝑁 ↑ 𝑋)) = (𝐴 × (𝑁 ↑ 𝑋))) |
36 | 35 | eqcomd 2824 | . . 3 ⊢ (𝜑 → (𝐴 × (𝑁 ↑ 𝑋)) = (((algSc‘𝑊)‘𝐴)(.r‘𝑊)(𝑁 ↑ 𝑋))) |
37 | 36 | fveq2d 6667 | . 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 20423 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑆)) |
41 | 1, 40 | syl 17 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑆)) |
42 | 2 | ply1lmod 20348 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod) |
43 | 14, 42 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
44 | 29, 30, 16, 43, 31, 22 | asclf 20039 | . . . 4 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶(Base‘𝑊)) |
45 | 44, 12 | ffvelrnd 6844 | . . 3 ⊢ (𝜑 → ((algSc‘𝑊)‘𝐴) ∈ (Base‘𝑊)) |
46 | evl1scvarpw.t2 | . . . 4 ⊢ ∙ = (.r‘𝑆) | |
47 | 22, 32, 46 | rhmmul 19408 | . . 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 1363 | . 2 ⊢ (𝜑 → (𝑄‘(((algSc‘𝑊)‘𝐴)(.r‘𝑊)(𝑁 ↑ 𝑋))) = ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋)))) |
49 | 38, 2, 6, 29 | evl1sca 20425 | . . . 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 20452 | . . . 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 6666 | . . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘(𝑅 ↑s 𝐵)) |
55 | 53, 54 | eqtri 2841 | . . . . . . . . 9 ⊢ 𝑀 = (mulGrp‘(𝑅 ↑s 𝐵)) |
56 | 55 | fveq2i 6666 | . . . . . . . 8 ⊢ (.g‘𝑀) = (.g‘(mulGrp‘(𝑅 ↑s 𝐵))) |
57 | 52, 56 | eqtri 2841 | . . . . . . 7 ⊢ 𝐹 = (.g‘(mulGrp‘(𝑅 ↑s 𝐵))) |
58 | 57 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (.g‘(mulGrp‘(𝑅 ↑s 𝐵)))) |
59 | 58 | eqcomd 2824 | . . . . 5 ⊢ (𝜑 → (.g‘(mulGrp‘(𝑅 ↑s 𝐵))) = 𝐹) |
60 | 59 | oveqd 7162 | . . . 4 ⊢ (𝜑 → (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))(𝑄‘𝑋)) = (𝑁𝐹(𝑄‘𝑋))) |
61 | 51, 60 | eqtrd 2853 | . . 3 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁𝐹(𝑄‘𝑋))) |
62 | 50, 61 | oveq12d 7163 | . 2 ⊢ (𝜑 → ((𝑄‘((algSc‘𝑊)‘𝐴)) ∙ (𝑄‘(𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋)))) |
63 | 37, 48, 62 | 3eqtrd 2857 | 1 ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {csn 4557 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ℕ0cn0 11885 Basecbs 16471 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 ↑s cpws 16708 Mndcmnd 17899 .gcmg 18162 mulGrpcmgp 19168 Ringcrg 19226 CRingccrg 19227 RingHom crh 19393 LModclmod 19563 AssAlgcasa 20010 algSccascl 20012 var1cv1 20272 Poly1cpl1 20273 eval1ce1 20405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-srg 19185 df-ring 19228 df-cring 19229 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-assa 20013 df-asp 20014 df-ascl 20015 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-evls 20214 df-evl 20215 df-psr1 20276 df-vr1 20277 df-ply1 20278 df-evls1 20406 df-evl1 20407 |
This theorem is referenced by: (None) |
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