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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 2734 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
4 | eqid 2734 | . . . 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 27288 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑍) ∖ (Unit‘𝑍)) × {0}) ⊆ 𝑥}) |
8 | 1, 2, 3, 4, 5, 7 | dchrval 27287 | . . 3 ⊢ (𝜑 → 𝐺 = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉}) |
9 | 8 | fveq2d 6923 | . 2 ⊢ (𝜑 → (+g‘𝐺) = (+g‘{〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉})) |
10 | dchrmul.t | . 2 ⊢ · = (+g‘𝐺) | |
11 | 6 | fvexi 6933 | . . . 4 ⊢ 𝐷 ∈ V |
12 | 11, 11 | xpex 7784 | . . 3 ⊢ (𝐷 × 𝐷) ∈ V |
13 | ofexg 7715 | . . 3 ⊢ ((𝐷 × 𝐷) ∈ V → ( ∘f · ↾ (𝐷 × 𝐷)) ∈ V) | |
14 | eqid 2734 | . . . 4 ⊢ {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉} = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉} | |
15 | 14 | grpplusg 17342 | . . 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 2799 | 1 ⊢ (𝜑 → · = ( ∘f · ↾ (𝐷 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 Vcvv 3482 {cpr 4650 〈cop 4654 × cxp 5697 ↾ cres 5701 ‘cfv 6572 ∘f cof 7708 · cmul 11185 ℕcn 12289 ndxcnx 17235 Basecbs 17253 +gcplusg 17306 Unitcui 20376 ℤ/nℤczn 21531 DChrcdchr 27285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-struct 17189 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-dchr 27286 |
This theorem is referenced by: dchrmul 27301 |
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