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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 2726 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
4 | eqid 2726 | . . . 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 27264 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑍) ∖ (Unit‘𝑍)) × {0}) ⊆ 𝑥}) |
8 | 1, 2, 3, 4, 5, 7 | dchrval 27263 | . . 3 ⊢ (𝜑 → 𝐺 = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉}) |
9 | 8 | fveq2d 6905 | . 2 ⊢ (𝜑 → (+g‘𝐺) = (+g‘{〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉})) |
10 | dchrmul.t | . 2 ⊢ · = (+g‘𝐺) | |
11 | 6 | fvexi 6915 | . . . 4 ⊢ 𝐷 ∈ V |
12 | 11, 11 | xpex 7761 | . . 3 ⊢ (𝐷 × 𝐷) ∈ V |
13 | ofexg 7695 | . . 3 ⊢ ((𝐷 × 𝐷) ∈ V → ( ∘f · ↾ (𝐷 × 𝐷)) ∈ V) | |
14 | eqid 2726 | . . . 4 ⊢ {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉} = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉} | |
15 | 14 | grpplusg 17302 | . . 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 2791 | 1 ⊢ (𝜑 → · = ( ∘f · ↾ (𝐷 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 {cpr 4635 〈cop 4639 × cxp 5680 ↾ cres 5684 ‘cfv 6554 ∘f cof 7688 · cmul 11163 ℕcn 12264 ndxcnx 17195 Basecbs 17213 +gcplusg 17266 Unitcui 20337 ℤ/nℤczn 21492 DChrcdchr 27261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-dchr 27262 |
This theorem is referenced by: dchrmul 27277 |
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