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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 25812 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 5 | nnnn0d 11949 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | dchrmhm.z | . . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
8 | 7 | zncrng 20684 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
9 | 6, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
10 | crngring 19301 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
12 | dchrelbas4.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
13 | 12 | zrhrhm 20652 | . . . . 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 20616 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
18 | zringmulr 20619 | . . . . 5 ⊢ · = (.r‘ℤring) | |
19 | eqid 2820 | . . . . 5 ⊢ (.r‘𝑍) = (.r‘𝑍) | |
20 | 17, 18, 19 | rhmmul 19472 | . . . 4 ⊢ ((𝐿 ∈ (ℤring RingHom 𝑍) ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) |
21 | 14, 15, 16, 20 | syl3anc 1366 | . . 3 ⊢ (𝜑 → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) |
22 | 21 | fveq2d 6667 | . 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶)))) |
23 | 2, 7, 3 | dchrmhm 25813 | . . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
24 | 23, 1 | sseldi 3958 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
25 | eqid 2820 | . . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
26 | 17, 25 | rhmf 19471 | . . . . 5 ⊢ (𝐿 ∈ (ℤring RingHom 𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) |
27 | 14, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
28 | 27, 15 | ffvelrnd 6845 | . . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍)) |
29 | 27, 16 | ffvelrnd 6845 | . . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍)) |
30 | eqid 2820 | . . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
31 | 30, 25 | mgpbas 19238 | . . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍)) |
32 | 30, 19 | mgpplusg 19236 | . . . 4 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
33 | eqid 2820 | . . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
34 | cnfldmul 20544 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
35 | 33, 34 | mgpplusg 19236 | . . . 4 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
36 | 31, 32, 35 | mhmlin 17956 | . . 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 1366 | . 2 ⊢ (𝜑 → (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
38 | 22, 37 | eqtrd 2855 | 1 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 · cmul 10535 ℕcn 11631 ℕ0cn0 11891 ℤcz 11975 Basecbs 16476 .rcmulr 16559 MndHom cmhm 17947 mulGrpcmgp 19232 Ringcrg 19290 CRingccrg 19291 RingHom crh 19457 ℂfldccnfld 20538 ℤringzring 20610 ℤRHomczrh 20640 ℤ/nℤczn 20643 DChrcdchr 25804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-ec 8284 df-qs 8288 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-seq 13367 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-starv 16573 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-unif 16581 df-0g 16708 df-imas 16774 df-qus 16775 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-mhm 17949 df-grp 18099 df-minusg 18100 df-sbg 18101 df-mulg 18218 df-subg 18269 df-nsg 18270 df-eqg 18271 df-ghm 18349 df-cmn 18901 df-abl 18902 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19366 df-rnghom 19460 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 df-2idl 19998 df-cnfld 20539 df-zring 20611 df-zrh 20644 df-zn 20647 df-dchr 25805 |
This theorem is referenced by: dchrmusum2 26066 dchrvmasumlem1 26067 dchrvmasum2lem 26068 dchrisum0fmul 26078 |
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