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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 24765 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 5 | nnnn0d 11228 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | dchrmhm.z | . . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
8 | 7 | zncrng 19712 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
9 | 6, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
10 | crngring 18381 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
12 | dchrelbas4.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
13 | 12 | zrhrhm 19679 | . . . . 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 19643 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
18 | zringmulr 19646 | . . . . 5 ⊢ · = (.r‘ℤring) | |
19 | eqid 2610 | . . . . 5 ⊢ (.r‘𝑍) = (.r‘𝑍) | |
20 | 17, 18, 19 | rhmmul 18550 | . . . 4 ⊢ ((𝐿 ∈ (ℤring RingHom 𝑍) ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) |
21 | 14, 15, 16, 20 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) |
22 | 21 | fveq2d 6107 | . 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶)))) |
23 | 2, 7, 3 | dchrmhm 24766 | . . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
24 | 23, 1 | sseldi 3566 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
25 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
26 | 17, 25 | rhmf 18549 | . . . . 5 ⊢ (𝐿 ∈ (ℤring RingHom 𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) |
27 | 14, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
28 | 27, 15 | ffvelrnd 6268 | . . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍)) |
29 | 27, 16 | ffvelrnd 6268 | . . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍)) |
30 | eqid 2610 | . . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
31 | 30, 25 | mgpbas 18318 | . . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍)) |
32 | 30, 19 | mgpplusg 18316 | . . . 4 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
33 | eqid 2610 | . . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
34 | cnfldmul 19573 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
35 | 33, 34 | mgpplusg 18316 | . . . 4 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
36 | 31, 32, 35 | mhmlin 17165 | . . 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 1318 | . 2 ⊢ (𝜑 → (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
38 | 22, 37 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 · cmul 9820 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 Basecbs 15695 .rcmulr 15769 MndHom cmhm 17156 mulGrpcmgp 18312 Ringcrg 18370 CRingccrg 18371 RingHom crh 18535 ℂfldccnfld 19567 ℤringzring 19637 ℤRHomczrh 19667 ℤ/nℤczn 19670 DChrcdchr 24757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-seq 12664 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 df-dchr 24758 |
This theorem is referenced by: dchrmusum2 24983 dchrvmasumlem1 24984 dchrvmasum2lem 24985 dchrisum0fmul 24995 |
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