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Mirrors > Home > MPE Home > Th. List > dchrmul | 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) |
dchrmul.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrmul.y | ⊢ (𝜑 → 𝑌 ∈ 𝐷) |
Ref | Expression |
---|---|
dchrmul | ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘𝑓 · 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrmhm.g | . . . 4 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchrmhm.z | . . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | dchrmhm.b | . . . 4 ⊢ 𝐷 = (Base‘𝐺) | |
4 | dchrmul.t | . . . 4 ⊢ · = (+g‘𝐺) | |
5 | dchrmul.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 1, 3 | dchrrcl 25314 | . . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
8 | 1, 2, 3, 4, 7 | dchrplusg 25321 | . . 3 ⊢ (𝜑 → · = ( ∘𝑓 · ↾ (𝐷 × 𝐷))) |
9 | 8 | oveqd 6893 | . 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋( ∘𝑓 · ↾ (𝐷 × 𝐷))𝑌)) |
10 | dchrmul.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
11 | 5, 10 | ofmresval 7142 | . 2 ⊢ (𝜑 → (𝑋( ∘𝑓 · ↾ (𝐷 × 𝐷))𝑌) = (𝑋 ∘𝑓 · 𝑌)) |
12 | 9, 11 | eqtrd 2831 | 1 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘𝑓 · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 × cxp 5308 ↾ cres 5312 ‘cfv 6099 (class class class)co 6876 ∘𝑓 cof 7127 · cmul 10227 ℕcn 11310 Basecbs 16181 +gcplusg 16264 ℤ/nℤczn 20170 DChrcdchr 25306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-plusg 16277 df-dchr 25307 |
This theorem is referenced by: dchrmulcl 25323 dchrmulid2 25326 dchrinvcl 25327 dchrabl 25328 dchrinv 25335 sumdchr2 25344 dchr2sum 25347 |
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