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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 | โข ๐ = (eval1โ๐ ) |
evl1varpw.w | โข ๐ = (Poly1โ๐ ) |
evl1varpw.g | โข ๐บ = (mulGrpโ๐) |
evl1varpw.x | โข ๐ = (var1โ๐ ) |
evl1varpw.b | โข ๐ต = (Baseโ๐ ) |
evl1varpw.e | โข โ = (.gโ๐บ) |
evl1varpw.r | โข (๐ โ ๐ โ CRing) |
evl1varpw.n | โข (๐ โ ๐ โ โ0) |
evl1scvarpw.t1 | โข ร = ( ยท๐ โ๐) |
evl1scvarpw.a | โข (๐ โ ๐ด โ ๐ต) |
evl1scvarpwval.c | โข (๐ โ ๐ถ โ ๐ต) |
evl1scvarpwval.h | โข ๐ป = (mulGrpโ๐ ) |
evl1scvarpwval.e | โข ๐ธ = (.gโ๐ป) |
evl1scvarpwval.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
evl1scvarpwval | โข (๐ โ ((๐โ(๐ด ร (๐ โ ๐)))โ๐ถ) = (๐ด ยท (๐๐ธ๐ถ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1varpw.q | . . 3 โข ๐ = (eval1โ๐ ) | |
2 | evl1varpw.w | . . 3 โข ๐ = (Poly1โ๐ ) | |
3 | evl1varpw.b | . . 3 โข ๐ต = (Baseโ๐ ) | |
4 | eqid 2727 | . . 3 โข (Baseโ๐) = (Baseโ๐) | |
5 | evl1varpw.r | . . 3 โข (๐ โ ๐ โ CRing) | |
6 | evl1scvarpwval.c | . . 3 โข (๐ โ ๐ถ โ ๐ต) | |
7 | evl1varpw.g | . . . . . 6 โข ๐บ = (mulGrpโ๐) | |
8 | 7, 4 | mgpbas 20071 | . . . . 5 โข (Baseโ๐) = (Baseโ๐บ) |
9 | evl1varpw.e | . . . . 5 โข โ = (.gโ๐บ) | |
10 | crngring 20176 | . . . . . . . 8 โข (๐ โ CRing โ ๐ โ Ring) | |
11 | 5, 10 | syl 17 | . . . . . . 7 โข (๐ โ ๐ โ Ring) |
12 | 2 | ply1ring 22153 | . . . . . . 7 โข (๐ โ Ring โ ๐ โ Ring) |
13 | 11, 12 | syl 17 | . . . . . 6 โข (๐ โ ๐ โ Ring) |
14 | 7 | ringmgp 20170 | . . . . . 6 โข (๐ โ Ring โ ๐บ โ Mnd) |
15 | 13, 14 | syl 17 | . . . . 5 โข (๐ โ ๐บ โ Mnd) |
16 | evl1varpw.n | . . . . 5 โข (๐ โ ๐ โ โ0) | |
17 | evl1varpw.x | . . . . . . 7 โข ๐ = (var1โ๐ ) | |
18 | 17, 2, 4 | vr1cl 22123 | . . . . . 6 โข (๐ โ Ring โ ๐ โ (Baseโ๐)) |
19 | 11, 18 | syl 17 | . . . . 5 โข (๐ โ ๐ โ (Baseโ๐)) |
20 | 8, 9, 15, 16, 19 | mulgnn0cld 19041 | . . . 4 โข (๐ โ (๐ โ ๐) โ (Baseโ๐)) |
21 | evl1scvarpwval.h | . . . . 5 โข ๐ป = (mulGrpโ๐ ) | |
22 | evl1scvarpwval.e | . . . . 5 โข ๐ธ = (.gโ๐ป) | |
23 | 1, 2, 7, 17, 3, 9, 5, 16, 6, 21, 22 | evl1varpwval 22268 | . . . 4 โข (๐ โ ((๐โ(๐ โ ๐))โ๐ถ) = (๐๐ธ๐ถ)) |
24 | 20, 23 | jca 511 | . . 3 โข (๐ โ ((๐ โ ๐) โ (Baseโ๐) โง ((๐โ(๐ โ ๐))โ๐ถ) = (๐๐ธ๐ถ))) |
25 | evl1scvarpw.a | . . 3 โข (๐ โ ๐ด โ ๐ต) | |
26 | evl1scvarpw.t1 | . . 3 โข ร = ( ยท๐ โ๐) | |
27 | evl1scvarpwval.t | . . 3 โข ยท = (.rโ๐ ) | |
28 | 1, 2, 3, 4, 5, 6, 24, 25, 26, 27 | evl1vsd 22250 | . 2 โข (๐ โ ((๐ด ร (๐ โ ๐)) โ (Baseโ๐) โง ((๐โ(๐ด ร (๐ โ ๐)))โ๐ถ) = (๐ด ยท (๐๐ธ๐ถ)))) |
29 | 28 | simprd 495 | 1 โข (๐ โ ((๐โ(๐ด ร (๐ โ ๐)))โ๐ถ) = (๐ด ยท (๐๐ธ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 โcfv 6542 (class class class)co 7414 โ0cn0 12494 Basecbs 17171 .rcmulr 17225 ยท๐ cvsca 17228 Mndcmnd 18685 .gcmg 19014 mulGrpcmgp 20065 Ringcrg 20164 CRingccrg 20165 var1cv1 22082 Poly1cpl1 22083 eval1ce1 22220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-srg 20118 df-ring 20166 df-cring 20167 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-lmod 20734 df-lss 20805 df-lsp 20845 df-assa 21774 df-asp 21775 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-evls 22005 df-evl 22006 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-evl1 22222 |
This theorem is referenced by: evl1gsummon 22271 |
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