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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 2825 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
4 | eqid 2825 | . . . 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 25380 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑍) ∖ (Unit‘𝑍)) × {0}) ⊆ 𝑥}) |
8 | 1, 2, 3, 4, 5, 7 | dchrval 25379 | . . 3 ⊢ (𝜑 → 𝐺 = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉}) |
9 | 8 | fveq2d 6441 | . 2 ⊢ (𝜑 → (+g‘𝐺) = (+g‘{〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉})) |
10 | dchrmul.t | . 2 ⊢ · = (+g‘𝐺) | |
11 | 6 | fvexi 6451 | . . . 4 ⊢ 𝐷 ∈ V |
12 | 11, 11 | xpex 7228 | . . 3 ⊢ (𝐷 × 𝐷) ∈ V |
13 | ofexg 7166 | . . 3 ⊢ ((𝐷 × 𝐷) ∈ V → ( ∘𝑓 · ↾ (𝐷 × 𝐷)) ∈ V) | |
14 | eqid 2825 | . . . 4 ⊢ {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉} = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉} | |
15 | 14 | grpplusg 16358 | . . 3 ⊢ (( ∘𝑓 · ↾ (𝐷 × 𝐷)) ∈ V → ( ∘𝑓 · ↾ (𝐷 × 𝐷)) = (+g‘{〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉})) |
16 | 12, 13, 15 | mp2b 10 | . 2 ⊢ ( ∘𝑓 · ↾ (𝐷 × 𝐷)) = (+g‘{〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))〉}) |
17 | 9, 10, 16 | 3eqtr4g 2886 | 1 ⊢ (𝜑 → · = ( ∘𝑓 · ↾ (𝐷 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 {cpr 4401 〈cop 4405 × cxp 5344 ↾ cres 5348 ‘cfv 6127 ∘𝑓 cof 7160 · cmul 10264 ℕcn 11357 ndxcnx 16226 Basecbs 16229 +gcplusg 16312 Unitcui 19000 ℤ/nℤczn 20218 DChrcdchr 25377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-plusg 16325 df-dchr 25378 |
This theorem is referenced by: dchrmul 25393 |
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