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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 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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