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Mirrors > Home > MPE Home > Th. List > dchrelbas | Structured version Visualization version GIF version |
Description: A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of ℂ, which is zero off the group of units of ℤ/nℤ. (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 |
---|---|
dchrelbas | ⊢ (𝜑 → (𝑋 ∈ 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 | dchrbas.b | . . . 4 ⊢ 𝐷 = (Base‘𝐺) | |
7 | 1, 2, 3, 4, 5, 6 | dchrbas 25813 | . . 3 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}) |
8 | 7 | eleq2d 2900 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ 𝑋 ∈ {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥})) |
9 | sseq2 3995 | . . 3 ⊢ (𝑥 = 𝑋 → (((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥 ↔ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑋)) | |
10 | 9 | elrab 3682 | . 2 ⊢ (𝑋 ∈ {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥} ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑋)) |
11 | 8, 10 | syl6bb 289 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 × cxp 5555 ‘cfv 6357 (class class class)co 7158 0cc0 10539 ℕcn 11640 Basecbs 16485 MndHom cmhm 17956 mulGrpcmgp 19241 Unitcui 19391 ℂfldccnfld 20547 ℤ/nℤczn 20652 DChrcdchr 25810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-dchr 25811 |
This theorem is referenced by: dchrelbas2 25815 dchrmhm 25819 |
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