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Mirrors > Home > MPE Home > Th. List > dchrplusg | Structured version Visualization version GIF version |
Description: Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrmhm.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrmhm.b | ⊢ 𝐷 = (Base‘𝐺) |
dchrmul.t | ⊢ · = (+g‘𝐺) |
dchrplusg.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
dchrplusg | ⊢ (𝜑 → · = ( ∘𝑓 · ↾ (𝐷 × 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrmhm.g | . . . 4 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchrmhm.z | . . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | eqid 2760 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
4 | eqid 2760 | . . . 4 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
5 | dchrplusg.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | dchrmhm.b | . . . . 5 ⊢ 𝐷 = (Base‘𝐺) | |
7 | 1, 2, 3, 4, 5, 6 | dchrbas 25159 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑍) ∖ (Unit‘𝑍)) × {0}) ⊆ 𝑥}) |
8 | 1, 2, 3, 4, 5, 7 | dchrval 25158 | . . 3 ⊢ (𝜑 → 𝐺 = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉}) |
9 | 8 | fveq2d 6356 | . 2 ⊢ (𝜑 → (+g‘𝐺) = (+g‘{〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉})) |
10 | dchrmul.t | . 2 ⊢ · = (+g‘𝐺) | |
11 | fvex 6362 | . . . . 5 ⊢ (Base‘𝐺) ∈ V | |
12 | 6, 11 | eqeltri 2835 | . . . 4 ⊢ 𝐷 ∈ V |
13 | 12, 12 | xpex 7127 | . . 3 ⊢ (𝐷 × 𝐷) ∈ V |
14 | ofexg 7066 | . . 3 ⊢ ((𝐷 × 𝐷) ∈ V → ( ∘𝑓 · ↾ (𝐷 × 𝐷)) ∈ V) | |
15 | eqid 2760 | . . . 4 ⊢ {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉} = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉} | |
16 | 15 | grpplusg 16194 | . . 3 ⊢ (( ∘𝑓 · ↾ (𝐷 × 𝐷)) ∈ V → ( ∘𝑓 · ↾ (𝐷 × 𝐷)) = (+g‘{〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉})) |
17 | 13, 14, 16 | mp2b 10 | . 2 ⊢ ( ∘𝑓 · ↾ (𝐷 × 𝐷)) = (+g‘{〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉}) |
18 | 9, 10, 17 | 3eqtr4g 2819 | 1 ⊢ (𝜑 → · = ( ∘𝑓 · ↾ (𝐷 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 Vcvv 3340 {cpr 4323 〈cop 4327 × cxp 5264 ↾ cres 5268 ‘cfv 6049 ∘𝑓 cof 7060 · cmul 10133 ℕcn 11212 ndxcnx 16056 Basecbs 16059 +gcplusg 16143 Unitcui 18839 ℤ/nℤczn 20053 DChrcdchr 25156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-plusg 16156 df-dchr 25157 |
This theorem is referenced by: dchrmul 25172 |
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