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Mirrors > Home > MPE Home > Th. List > evl1varpw | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 21720, the proof is shorter using evls1varpw 21689 instead of proving it directly. (Contributed by AV, 15-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) |
Ref | Expression |
---|---|
evl1varpw | ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))(𝑄‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1varpw.q | . . . . 5 ⊢ 𝑄 = (eval1‘𝑅) | |
2 | evl1varpw.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 1, 2 | evl1fval1 21693 | . . . 4 ⊢ 𝑄 = (𝑅 evalSub1 𝐵) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑄 = (𝑅 evalSub1 𝐵)) |
5 | evl1varpw.e | . . . . . 6 ⊢ ↑ = (.g‘𝐺) | |
6 | evl1varpw.g | . . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊) | |
7 | evl1varpw.w | . . . . . . . . 9 ⊢ 𝑊 = (Poly1‘𝑅) | |
8 | 7 | fveq2i 6843 | . . . . . . . 8 ⊢ (mulGrp‘𝑊) = (mulGrp‘(Poly1‘𝑅)) |
9 | 6, 8 | eqtri 2764 | . . . . . . 7 ⊢ 𝐺 = (mulGrp‘(Poly1‘𝑅)) |
10 | 9 | fveq2i 6843 | . . . . . 6 ⊢ (.g‘𝐺) = (.g‘(mulGrp‘(Poly1‘𝑅))) |
11 | 5, 10 | eqtri 2764 | . . . . 5 ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝑅))) |
12 | evl1varpw.r | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
13 | 2 | ressid 17122 | . . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
14 | 12, 13 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝑅 ↾s 𝐵) = 𝑅) |
15 | 14 | eqcomd 2742 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 = (𝑅 ↾s 𝐵)) |
16 | 15 | fveq2d 6844 | . . . . . . 7 ⊢ (𝜑 → (Poly1‘𝑅) = (Poly1‘(𝑅 ↾s 𝐵))) |
17 | 16 | fveq2d 6844 | . . . . . 6 ⊢ (𝜑 → (mulGrp‘(Poly1‘𝑅)) = (mulGrp‘(Poly1‘(𝑅 ↾s 𝐵)))) |
18 | 17 | fveq2d 6844 | . . . . 5 ⊢ (𝜑 → (.g‘(mulGrp‘(Poly1‘𝑅))) = (.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))))) |
19 | 11, 18 | eqtrid 2788 | . . . 4 ⊢ (𝜑 → ↑ = (.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))))) |
20 | eqidd 2737 | . . . 4 ⊢ (𝜑 → 𝑁 = 𝑁) | |
21 | evl1varpw.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
22 | 15 | fveq2d 6844 | . . . . 5 ⊢ (𝜑 → (var1‘𝑅) = (var1‘(𝑅 ↾s 𝐵))) |
23 | 21, 22 | eqtrid 2788 | . . . 4 ⊢ (𝜑 → 𝑋 = (var1‘(𝑅 ↾s 𝐵))) |
24 | 19, 20, 23 | oveq123d 7375 | . . 3 ⊢ (𝜑 → (𝑁 ↑ 𝑋) = (𝑁(.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))))(var1‘(𝑅 ↾s 𝐵)))) |
25 | 4, 24 | fveq12d 6847 | . 2 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = ((𝑅 evalSub1 𝐵)‘(𝑁(.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))))(var1‘(𝑅 ↾s 𝐵))))) |
26 | eqid 2736 | . . 3 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵) | |
27 | eqid 2736 | . . 3 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
28 | eqid 2736 | . . 3 ⊢ (Poly1‘(𝑅 ↾s 𝐵)) = (Poly1‘(𝑅 ↾s 𝐵)) | |
29 | eqid 2736 | . . 3 ⊢ (mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))) = (mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))) | |
30 | eqid 2736 | . . 3 ⊢ (var1‘(𝑅 ↾s 𝐵)) = (var1‘(𝑅 ↾s 𝐵)) | |
31 | eqid 2736 | . . 3 ⊢ (.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵)))) = (.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵)))) | |
32 | crngring 19972 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
33 | 2 | subrgid 20220 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
34 | 12, 32, 33 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅)) |
35 | evl1varpw.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
36 | 26, 27, 28, 29, 30, 2, 31, 12, 34, 35 | evls1varpw 21689 | . 2 ⊢ (𝜑 → ((𝑅 evalSub1 𝐵)‘(𝑁(.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))))(var1‘(𝑅 ↾s 𝐵)))) = (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))((𝑅 evalSub1 𝐵)‘(var1‘(𝑅 ↾s 𝐵))))) |
37 | 3 | eqcomi 2745 | . . . . 5 ⊢ (𝑅 evalSub1 𝐵) = 𝑄 |
38 | 37 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑅 evalSub1 𝐵) = 𝑄) |
39 | 23 | eqcomd 2742 | . . . 4 ⊢ (𝜑 → (var1‘(𝑅 ↾s 𝐵)) = 𝑋) |
40 | 38, 39 | fveq12d 6847 | . . 3 ⊢ (𝜑 → ((𝑅 evalSub1 𝐵)‘(var1‘(𝑅 ↾s 𝐵))) = (𝑄‘𝑋)) |
41 | 40 | oveq2d 7370 | . 2 ⊢ (𝜑 → (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))((𝑅 evalSub1 𝐵)‘(var1‘(𝑅 ↾s 𝐵)))) = (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))(𝑄‘𝑋))) |
42 | 25, 36, 41 | 3eqtrd 2780 | 1 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))(𝑄‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6494 (class class class)co 7354 ℕ0cn0 12410 Basecbs 17080 ↾s cress 17109 ↑s cpws 17325 .gcmg 18868 mulGrpcmgp 19892 Ringcrg 19960 CRingccrg 19961 SubRingcsubrg 20214 var1cv1 21543 Poly1cpl1 21544 evalSub1 ces1 21675 eval1ce1 21676 |
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 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7614 df-ofr 7615 df-om 7800 df-1st 7918 df-2nd 7919 df-supp 8090 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-map 8764 df-pm 8765 df-ixp 8833 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-fsupp 9303 df-sup 9375 df-oi 9443 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-uz 12761 df-fz 13422 df-fzo 13565 df-seq 13904 df-hash 14228 df-struct 17016 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-sca 17146 df-vsca 17147 df-ip 17148 df-tset 17149 df-ple 17150 df-ds 17152 df-hom 17154 df-cco 17155 df-0g 17320 df-gsum 17321 df-prds 17326 df-pws 17328 df-mre 17463 df-mrc 17464 df-acs 17466 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-mhm 18598 df-submnd 18599 df-grp 18748 df-minusg 18749 df-sbg 18750 df-mulg 18869 df-subg 18921 df-ghm 19002 df-cntz 19093 df-cmn 19560 df-abl 19561 df-mgp 19893 df-ur 19910 df-srg 19914 df-ring 19962 df-cring 19963 df-rnghom 20142 df-subrg 20216 df-lmod 20320 df-lss 20389 df-lsp 20429 df-assa 21255 df-asp 21256 df-ascl 21257 df-psr 21307 df-mvr 21308 df-mpl 21309 df-opsr 21311 df-evls 21478 df-evl 21479 df-psr1 21547 df-vr1 21548 df-ply1 21549 df-evls1 21677 df-evl1 21678 |
This theorem is referenced by: evl1scvarpw 21725 |
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