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Mirrors > Home > MPE Home > Th. List > dchrbas | Structured version Visualization version GIF version |
Description: Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.) |
Ref | Expression |
---|---|
dchrval.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrval.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrval.b | ⊢ 𝐵 = (Base‘𝑍) |
dchrval.u | ⊢ 𝑈 = (Unit‘𝑍) |
dchrval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
dchrbas.b | ⊢ 𝐷 = (Base‘𝐺) |
Ref | Expression |
---|---|
dchrbas | ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrval.g | . . . 4 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchrval.z | . . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | dchrval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑍) | |
4 | dchrval.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑍) | |
5 | dchrval.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | eqidd 2774 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}) | |
7 | 1, 2, 3, 4, 5, 6 | dchrval 25528 | . . 3 ⊢ (𝜑 → 𝐺 = {〈(Base‘ndx), {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ ({𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} × {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}))〉}) |
8 | 7 | fveq2d 6501 | . 2 ⊢ (𝜑 → (Base‘𝐺) = (Base‘{〈(Base‘ndx), {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ ({𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} × {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}))〉})) |
9 | dchrbas.b | . 2 ⊢ 𝐷 = (Base‘𝐺) | |
10 | ovex 7007 | . . . 4 ⊢ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∈ V | |
11 | 10 | rabex 5088 | . . 3 ⊢ {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} ∈ V |
12 | eqid 2773 | . . . 4 ⊢ {〈(Base‘ndx), {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ ({𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} × {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}))〉} = {〈(Base‘ndx), {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ ({𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} × {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}))〉} | |
13 | 12 | grpbase 16465 | . . 3 ⊢ ({𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} ∈ V → {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} = (Base‘{〈(Base‘ndx), {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ ({𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} × {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}))〉})) |
14 | 11, 13 | ax-mp 5 | . 2 ⊢ {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} = (Base‘{〈(Base‘ndx), {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}〉, 〈(+g‘ndx), ( ∘𝑓 · ↾ ({𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} × {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}))〉}) |
15 | 8, 9, 14 | 3eqtr4g 2834 | 1 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 {crab 3087 Vcvv 3410 ∖ cdif 3821 ⊆ wss 3824 {csn 4436 {cpr 4438 〈cop 4442 × cxp 5402 ↾ cres 5406 ‘cfv 6186 (class class class)co 6975 ∘𝑓 cof 7224 0cc0 10334 · cmul 10339 ℕcn 11438 ndxcnx 16335 Basecbs 16338 +gcplusg 16420 MndHom cmhm 17814 mulGrpcmgp 18975 Unitcui 19125 ℂfldccnfld 20263 ℤ/nℤczn 20368 DChrcdchr 25526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-oadd 7908 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-struct 16340 df-ndx 16341 df-slot 16342 df-base 16344 df-plusg 16433 df-dchr 25527 |
This theorem is referenced by: dchrelbas 25530 dchrplusg 25541 |
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