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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evls1varpwval | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 21679. (Contributed by Thierry Arnoux, 24-Jan-2025.) |
Ref | Expression |
---|---|
evls1varpwval.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1varpwval.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1varpwval.w | ⊢ 𝑊 = (Poly1‘𝑈) |
evls1varpwval.x | ⊢ 𝑋 = (var1‘𝑈) |
evls1varpwval.b | ⊢ 𝐵 = (Base‘𝑆) |
evls1varpwval.e | ⊢ ∧ = (.g‘(mulGrp‘𝑊)) |
evls1varpwval.f | ⊢ ↑ = (.g‘(mulGrp‘𝑆)) |
evls1varpwval.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1varpwval.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evls1varpwval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
evls1varpwval.c | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
Ref | Expression |
---|---|
evls1varpwval | ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑋))‘𝐶) = (𝑁 ↑ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1varpwval.q | . . 3 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
2 | evls1varpwval.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
3 | evls1varpwval.w | . . 3 ⊢ 𝑊 = (Poly1‘𝑈) | |
4 | evls1varpwval.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
5 | eqid 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | evls1varpwval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
7 | evls1varpwval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
8 | evls1varpwval.e | . . 3 ⊢ ∧ = (.g‘(mulGrp‘𝑊)) | |
9 | evls1varpwval.f | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑆)) | |
10 | evls1varpwval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
11 | 4 | subrgring 20177 | . . . 4 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
12 | evls1varpwval.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑈) | |
13 | 12, 3, 5 | vr1cl 21539 | . . . 4 ⊢ (𝑈 ∈ Ring → 𝑋 ∈ (Base‘𝑊)) |
14 | 7, 11, 13 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
15 | evls1varpwval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15 | evls1expd 32086 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑋))‘𝐶) = (𝑁 ↑ ((𝑄‘𝑋)‘𝐶))) |
17 | 1, 12, 4, 2, 6, 7 | evls1var 21655 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵)) |
18 | 17 | fveq1d 6841 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑋)‘𝐶) = (( I ↾ 𝐵)‘𝐶)) |
19 | fvresi 7115 | . . . . 5 ⊢ (𝐶 ∈ 𝐵 → (( I ↾ 𝐵)‘𝐶) = 𝐶) | |
20 | 15, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐵)‘𝐶) = 𝐶) |
21 | 18, 20 | eqtrd 2777 | . . 3 ⊢ (𝜑 → ((𝑄‘𝑋)‘𝐶) = 𝐶) |
22 | 21 | oveq2d 7367 | . 2 ⊢ (𝜑 → (𝑁 ↑ ((𝑄‘𝑋)‘𝐶)) = (𝑁 ↑ 𝐶)) |
23 | 16, 22 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑋))‘𝐶) = (𝑁 ↑ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 I cid 5528 ↾ cres 5633 ‘cfv 6493 (class class class)co 7351 ℕ0cn0 12371 Basecbs 17042 ↾s cress 17071 .gcmg 18830 mulGrpcmgp 19854 Ringcrg 19917 CRingccrg 19918 SubRingcsubrg 20170 var1cv1 21498 Poly1cpl1 21499 evalSub1 ces1 21630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-ofr 7610 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14184 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-hom 17116 df-cco 17117 df-0g 17282 df-gsum 17283 df-prds 17288 df-pws 17290 df-mre 17425 df-mrc 17426 df-acs 17428 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-mhm 18560 df-submnd 18561 df-grp 18710 df-minusg 18711 df-sbg 18712 df-mulg 18831 df-subg 18883 df-ghm 18964 df-cntz 19055 df-cmn 19522 df-abl 19523 df-mgp 19855 df-ur 19872 df-srg 19876 df-ring 19919 df-cring 19920 df-rnghom 20098 df-subrg 20172 df-lmod 20276 df-lss 20345 df-lsp 20385 df-assa 21211 df-asp 21212 df-ascl 21213 df-psr 21263 df-mvr 21264 df-mpl 21265 df-opsr 21267 df-evls 21433 df-evl 21434 df-psr1 21502 df-vr1 21503 df-ply1 21504 df-evls1 21632 df-evl1 21633 |
This theorem is referenced by: evls1fpws 32088 |
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