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Mirrors > Home > MPE Home > Th. List > evl1scvarpwval | Structured version Visualization version GIF version |
Description: Value of a univariate polynomial evaluation mapping a multiple of an exponentiation of a variable to the multiple of the 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) |
evl1scvarpwval.c | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
evl1scvarpwval.h | ⊢ 𝐻 = (mulGrp‘𝑅) |
evl1scvarpwval.e | ⊢ 𝐸 = (.g‘𝐻) |
evl1scvarpwval.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
evl1scvarpwval | ⊢ (𝜑 → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1varpw.q | . . 3 ⊢ 𝑄 = (eval1‘𝑅) | |
2 | evl1varpw.w | . . 3 ⊢ 𝑊 = (Poly1‘𝑅) | |
3 | evl1varpw.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | eqid 2778 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | evl1varpw.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
6 | evl1scvarpwval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
7 | crngring 19031 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | 5, 7 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | 2 | ply1ring 20119 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring) |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Ring) |
11 | evl1varpw.g | . . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑊) | |
12 | 11 | ringmgp 19026 | . . . . . 6 ⊢ (𝑊 ∈ Ring → 𝐺 ∈ Mnd) |
13 | 10, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
14 | evl1varpw.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
15 | evl1varpw.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
16 | 15, 2, 4 | vr1cl 20088 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑊)) |
17 | 8, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
18 | 11, 4 | mgpbas 18968 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝐺) |
19 | evl1varpw.e | . . . . . 6 ⊢ ↑ = (.g‘𝐺) | |
20 | 18, 19 | mulgnn0cl 18029 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) |
21 | 13, 14, 17, 20 | syl3anc 1351 | . . . 4 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑊)) |
22 | evl1scvarpwval.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑅) | |
23 | evl1scvarpwval.e | . . . . 5 ⊢ 𝐸 = (.g‘𝐻) | |
24 | 1, 2, 11, 15, 3, 19, 5, 14, 6, 22, 23 | evl1varpwval 20227 | . . . 4 ⊢ (𝜑 → ((𝑄‘(𝑁 ↑ 𝑋))‘𝐶) = (𝑁𝐸𝐶)) |
25 | 21, 24 | jca 504 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝑋) ∈ (Base‘𝑊) ∧ ((𝑄‘(𝑁 ↑ 𝑋))‘𝐶) = (𝑁𝐸𝐶))) |
26 | evl1scvarpw.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
27 | evl1scvarpw.t1 | . . 3 ⊢ × = ( ·𝑠 ‘𝑊) | |
28 | evl1scvarpwval.t | . . 3 ⊢ · = (.r‘𝑅) | |
29 | 1, 2, 3, 4, 5, 6, 25, 26, 27, 28 | evl1vsd 20209 | . 2 ⊢ (𝜑 → ((𝐴 × (𝑁 ↑ 𝑋)) ∈ (Base‘𝑊) ∧ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶)))) |
30 | 29 | simprd 488 | 1 ⊢ (𝜑 → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 ℕ0cn0 11707 Basecbs 16339 .rcmulr 16422 ·𝑠 cvsca 16425 Mndcmnd 17762 .gcmg 18011 mulGrpcmgp 18962 Ringcrg 19020 CRingccrg 19021 var1cv1 20047 Poly1cpl1 20048 eval1ce1 20180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-ofr 7228 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-fzo 12850 df-seq 13185 df-hash 13506 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-hom 16445 df-cco 16446 df-0g 16571 df-gsum 16572 df-prds 16577 df-pws 16579 df-mre 16715 df-mrc 16716 df-acs 16718 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-ghm 18127 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-srg 18979 df-ring 19022 df-cring 19023 df-rnghom 19190 df-subrg 19256 df-lmod 19358 df-lss 19426 df-lsp 19466 df-assa 19806 df-asp 19807 df-ascl 19808 df-psr 19850 df-mvr 19851 df-mpl 19852 df-opsr 19854 df-evls 19999 df-evl 20000 df-psr1 20051 df-vr1 20052 df-ply1 20053 df-evl1 20182 |
This theorem is referenced by: evl1gsummon 20230 |
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