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Mirrors > Home > MPE Home > Th. List > dchrplusg | Structured version Visualization version GIF version |
Description: Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | β’ πΊ = (DChrβπ) |
dchrmhm.z | β’ π = (β€/nβ€βπ) |
dchrmhm.b | β’ π· = (BaseβπΊ) |
dchrmul.t | β’ Β· = (+gβπΊ) |
dchrplusg.n | β’ (π β π β β) |
Ref | Expression |
---|---|
dchrplusg | β’ (π β Β· = ( βf Β· βΎ (π· Γ π·))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrmhm.g | . . . 4 β’ πΊ = (DChrβπ) | |
2 | dchrmhm.z | . . . 4 β’ π = (β€/nβ€βπ) | |
3 | eqid 2726 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
4 | eqid 2726 | . . . 4 β’ (Unitβπ) = (Unitβπ) | |
5 | dchrplusg.n | . . . 4 β’ (π β π β β) | |
6 | dchrmhm.b | . . . . 5 β’ π· = (BaseβπΊ) | |
7 | 1, 2, 3, 4, 5, 6 | dchrbas 27123 | . . . 4 β’ (π β π· = {π₯ β ((mulGrpβπ) MndHom (mulGrpββfld)) β£ (((Baseβπ) β (Unitβπ)) Γ {0}) β π₯}) |
8 | 1, 2, 3, 4, 5, 7 | dchrval 27122 | . . 3 β’ (π β πΊ = {β¨(Baseβndx), π·β©, β¨(+gβndx), ( βf Β· βΎ (π· Γ π·))β©}) |
9 | 8 | fveq2d 6889 | . 2 β’ (π β (+gβπΊ) = (+gβ{β¨(Baseβndx), π·β©, β¨(+gβndx), ( βf Β· βΎ (π· Γ π·))β©})) |
10 | dchrmul.t | . 2 β’ Β· = (+gβπΊ) | |
11 | 6 | fvexi 6899 | . . . 4 β’ π· β V |
12 | 11, 11 | xpex 7737 | . . 3 β’ (π· Γ π·) β V |
13 | ofexg 7672 | . . 3 β’ ((π· Γ π·) β V β ( βf Β· βΎ (π· Γ π·)) β V) | |
14 | eqid 2726 | . . . 4 β’ {β¨(Baseβndx), π·β©, β¨(+gβndx), ( βf Β· βΎ (π· Γ π·))β©} = {β¨(Baseβndx), π·β©, β¨(+gβndx), ( βf Β· βΎ (π· Γ π·))β©} | |
15 | 14 | grpplusg 17242 | . . 3 β’ (( βf Β· βΎ (π· Γ π·)) β V β ( βf Β· βΎ (π· Γ π·)) = (+gβ{β¨(Baseβndx), π·β©, β¨(+gβndx), ( βf Β· βΎ (π· Γ π·))β©})) |
16 | 12, 13, 15 | mp2b 10 | . 2 β’ ( βf Β· βΎ (π· Γ π·)) = (+gβ{β¨(Baseβndx), π·β©, β¨(+gβndx), ( βf Β· βΎ (π· Γ π·))β©}) |
17 | 9, 10, 16 | 3eqtr4g 2791 | 1 β’ (π β Β· = ( βf Β· βΎ (π· Γ π·))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 {cpr 4625 β¨cop 4629 Γ cxp 5667 βΎ cres 5671 βcfv 6537 βf cof 7665 Β· cmul 11117 βcn 12216 ndxcnx 17135 Basecbs 17153 +gcplusg 17206 Unitcui 20257 β€/nβ€czn 21389 DChrcdchr 27120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-dchr 27121 |
This theorem is referenced by: dchrmul 27136 |
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