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Mirrors > Home > MPE Home > Th. List > dchrzrhmul | Structured version Visualization version GIF version |
Description: A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrmhm.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrmhm.b | ⊢ 𝐷 = (Base‘𝐺) |
dchrelbas4.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
dchrzrh1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrzrh1.a | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
dchrzrh1.c | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
Ref | Expression |
---|---|
dchrzrhmul | ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrzrh1.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
2 | dchrmhm.g | . . . . . . . . . 10 ⊢ 𝐺 = (DChr‘𝑁) | |
3 | dchrmhm.b | . . . . . . . . . 10 ⊢ 𝐷 = (Base‘𝐺) | |
4 | 2, 3 | dchrrcl 25816 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 5 | nnnn0d 11956 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | dchrmhm.z | . . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
8 | 7 | zncrng 20691 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
9 | 6, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
10 | crngring 19308 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
12 | dchrelbas4.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
13 | 12 | zrhrhm 20659 | . . . . 5 ⊢ (𝑍 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑍)) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom 𝑍)) |
15 | dchrzrh1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
16 | dchrzrh1.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
17 | zringbas 20623 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
18 | zringmulr 20626 | . . . . 5 ⊢ · = (.r‘ℤring) | |
19 | eqid 2821 | . . . . 5 ⊢ (.r‘𝑍) = (.r‘𝑍) | |
20 | 17, 18, 19 | rhmmul 19479 | . . . 4 ⊢ ((𝐿 ∈ (ℤring RingHom 𝑍) ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) |
21 | 14, 15, 16, 20 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) |
22 | 21 | fveq2d 6674 | . 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶)))) |
23 | 2, 7, 3 | dchrmhm 25817 | . . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
24 | 23, 1 | sseldi 3965 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
25 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
26 | 17, 25 | rhmf 19478 | . . . . 5 ⊢ (𝐿 ∈ (ℤring RingHom 𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) |
27 | 14, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
28 | 27, 15 | ffvelrnd 6852 | . . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍)) |
29 | 27, 16 | ffvelrnd 6852 | . . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍)) |
30 | eqid 2821 | . . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
31 | 30, 25 | mgpbas 19245 | . . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍)) |
32 | 30, 19 | mgpplusg 19243 | . . . 4 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
33 | eqid 2821 | . . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
34 | cnfldmul 20551 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
35 | 33, 34 | mgpplusg 19243 | . . . 4 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
36 | 31, 32, 35 | mhmlin 17963 | . . 3 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝐿‘𝐴) ∈ (Base‘𝑍) ∧ (𝐿‘𝐶) ∈ (Base‘𝑍)) → (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
37 | 24, 28, 29, 36 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
38 | 22, 37 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 · cmul 10542 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 Basecbs 16483 .rcmulr 16566 MndHom cmhm 17954 mulGrpcmgp 19239 Ringcrg 19297 CRingccrg 19298 RingHom crh 19464 ℂfldccnfld 20545 ℤringzring 20617 ℤRHomczrh 20647 ℤ/nℤczn 20650 DChrcdchr 25808 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-ec 8291 df-qs 8295 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-seq 13371 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-imas 16781 df-qus 16782 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-nsg 18277 df-eqg 18278 df-ghm 18356 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-rnghom 19467 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-sra 19944 df-rgmod 19945 df-lidl 19946 df-rsp 19947 df-2idl 20005 df-cnfld 20546 df-zring 20618 df-zrh 20651 df-zn 20654 df-dchr 25809 |
This theorem is referenced by: dchrmusum2 26070 dchrvmasumlem1 26071 dchrvmasum2lem 26072 dchrisum0fmul 26082 |
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