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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 21228, the proof is shorter using evls1varpw 21197 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 21201 | . . . 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 6698 | . . . . . . . 8 ⊢ (mulGrp‘𝑊) = (mulGrp‘(Poly1‘𝑅)) |
9 | 6, 8 | eqtri 2759 | . . . . . . 7 ⊢ 𝐺 = (mulGrp‘(Poly1‘𝑅)) |
10 | 9 | fveq2i 6698 | . . . . . 6 ⊢ (.g‘𝐺) = (.g‘(mulGrp‘(Poly1‘𝑅))) |
11 | 5, 10 | eqtri 2759 | . . . . 5 ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝑅))) |
12 | evl1varpw.r | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
13 | 2 | ressid 16743 | . . . . . . . . . 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 6699 | . . . . . . 7 ⊢ (𝜑 → (Poly1‘𝑅) = (Poly1‘(𝑅 ↾s 𝐵))) |
17 | 16 | fveq2d 6699 | . . . . . 6 ⊢ (𝜑 → (mulGrp‘(Poly1‘𝑅)) = (mulGrp‘(Poly1‘(𝑅 ↾s 𝐵)))) |
18 | 17 | fveq2d 6699 | . . . . 5 ⊢ (𝜑 → (.g‘(mulGrp‘(Poly1‘𝑅))) = (.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))))) |
19 | 11, 18 | syl5eq 2783 | . . . 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 6699 | . . . . 5 ⊢ (𝜑 → (var1‘𝑅) = (var1‘(𝑅 ↾s 𝐵))) |
23 | 21, 22 | syl5eq 2783 | . . . 4 ⊢ (𝜑 → 𝑋 = (var1‘(𝑅 ↾s 𝐵))) |
24 | 19, 20, 23 | oveq123d 7212 | . . 3 ⊢ (𝜑 → (𝑁 ↑ 𝑋) = (𝑁(.g‘(mulGrp‘(Poly1‘(𝑅 ↾s 𝐵))))(var1‘(𝑅 ↾s 𝐵)))) |
25 | 4, 24 | fveq12d 6702 | . 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 19528 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
33 | 2 | subrgid 19756 | . . . 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 21197 | . 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 6702 | . . 3 ⊢ (𝜑 → ((𝑅 evalSub1 𝐵)‘(var1‘(𝑅 ↾s 𝐵))) = (𝑄‘𝑋)) |
41 | 40 | oveq2d 7207 | . 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 2775 | 1 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))(𝑄‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 ℕ0cn0 12055 Basecbs 16666 ↾s cress 16667 ↑s cpws 16905 .gcmg 18442 mulGrpcmgp 19458 Ringcrg 19516 CRingccrg 19517 SubRingcsubrg 19750 var1cv1 21051 Poly1cpl1 21052 evalSub1 ces1 21183 eval1ce1 21184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-0g 16900 df-gsum 16901 df-prds 16906 df-pws 16908 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-srg 19475 df-ring 19518 df-cring 19519 df-rnghom 19689 df-subrg 19752 df-lmod 19855 df-lss 19923 df-lsp 19963 df-assa 20769 df-asp 20770 df-ascl 20771 df-psr 20822 df-mvr 20823 df-mpl 20824 df-opsr 20826 df-evls 20986 df-evl 20987 df-psr1 21055 df-vr1 21056 df-ply1 21057 df-evls1 21185 df-evl1 21186 |
This theorem is referenced by: evl1scvarpw 21233 |
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