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Theorem dchrfi 26758
Description: The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
dchrabl.g 𝐺 = (DChrβ€˜π‘)
dchrfi.b 𝐷 = (Baseβ€˜πΊ)
Assertion
Ref Expression
dchrfi (𝑁 ∈ β„• β†’ 𝐷 ∈ Fin)

Proof of Theorem dchrfi
Dummy variables π‘₯ 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snfi 9044 . . . 4 {0} ∈ Fin
2 cnex 11191 . . . . . . . . 9 β„‚ ∈ V
32a1i 11 . . . . . . . 8 (𝑁 ∈ β„• β†’ β„‚ ∈ V)
4 ovexd 7444 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑧 ∈ β„‚) β†’ (𝑧↑(Ο•β€˜π‘)) ∈ V)
5 1cnd 11209 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑧 ∈ β„‚) β†’ 1 ∈ β„‚)
6 eqidd 2734 . . . . . . . 8 (𝑁 ∈ β„• β†’ (𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) = (𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))))
7 fconstmpt 5739 . . . . . . . . 9 (β„‚ Γ— {1}) = (𝑧 ∈ β„‚ ↦ 1)
87a1i 11 . . . . . . . 8 (𝑁 ∈ β„• β†’ (β„‚ Γ— {1}) = (𝑧 ∈ β„‚ ↦ 1))
93, 4, 5, 6, 8offval2 7690 . . . . . . 7 (𝑁 ∈ β„• β†’ ((𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∘f βˆ’ (β„‚ Γ— {1})) = (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)))
10 ssid 4005 . . . . . . . . . 10 β„‚ βŠ† β„‚
1110a1i 11 . . . . . . . . 9 (𝑁 ∈ β„• β†’ β„‚ βŠ† β„‚)
12 1cnd 11209 . . . . . . . . 9 (𝑁 ∈ β„• β†’ 1 ∈ β„‚)
13 phicl 16702 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (Ο•β€˜π‘) ∈ β„•)
1413nnnn0d 12532 . . . . . . . . 9 (𝑁 ∈ β„• β†’ (Ο•β€˜π‘) ∈ β„•0)
15 plypow 25719 . . . . . . . . 9 ((β„‚ βŠ† β„‚ ∧ 1 ∈ β„‚ ∧ (Ο•β€˜π‘) ∈ β„•0) β†’ (𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∈ (Polyβ€˜β„‚))
1611, 12, 14, 15syl3anc 1372 . . . . . . . 8 (𝑁 ∈ β„• β†’ (𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∈ (Polyβ€˜β„‚))
17 ax-1cn 11168 . . . . . . . . 9 1 ∈ β„‚
18 plyconst 25720 . . . . . . . . 9 ((β„‚ βŠ† β„‚ ∧ 1 ∈ β„‚) β†’ (β„‚ Γ— {1}) ∈ (Polyβ€˜β„‚))
1910, 17, 18mp2an 691 . . . . . . . 8 (β„‚ Γ— {1}) ∈ (Polyβ€˜β„‚)
20 plysubcl 25736 . . . . . . . 8 (((𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∈ (Polyβ€˜β„‚) ∧ (β„‚ Γ— {1}) ∈ (Polyβ€˜β„‚)) β†’ ((𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∘f βˆ’ (β„‚ Γ— {1})) ∈ (Polyβ€˜β„‚))
2116, 19, 20sylancl 587 . . . . . . 7 (𝑁 ∈ β„• β†’ ((𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∘f βˆ’ (β„‚ Γ— {1})) ∈ (Polyβ€˜β„‚))
229, 21eqeltrrd 2835 . . . . . 6 (𝑁 ∈ β„• β†’ (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) ∈ (Polyβ€˜β„‚))
23 0cn 11206 . . . . . . 7 0 ∈ β„‚
24 neg1ne0 12328 . . . . . . . 8 -1 β‰  0
25130expd 14104 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ (0↑(Ο•β€˜π‘)) = 0)
2625oveq1d 7424 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ ((0↑(Ο•β€˜π‘)) βˆ’ 1) = (0 βˆ’ 1))
27 oveq1 7416 . . . . . . . . . . . . 13 (𝑧 = 0 β†’ (𝑧↑(Ο•β€˜π‘)) = (0↑(Ο•β€˜π‘)))
2827oveq1d 7424 . . . . . . . . . . . 12 (𝑧 = 0 β†’ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = ((0↑(Ο•β€˜π‘)) βˆ’ 1))
29 eqid 2733 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) = (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))
30 ovex 7442 . . . . . . . . . . . 12 ((0↑(Ο•β€˜π‘)) βˆ’ 1) ∈ V
3128, 29, 30fvmpt 6999 . . . . . . . . . . 11 (0 ∈ β„‚ β†’ ((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) = ((0↑(Ο•β€˜π‘)) βˆ’ 1))
3223, 31ax-mp 5 . . . . . . . . . 10 ((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) = ((0↑(Ο•β€˜π‘)) βˆ’ 1)
33 df-neg 11447 . . . . . . . . . 10 -1 = (0 βˆ’ 1)
3426, 32, 333eqtr4g 2798 . . . . . . . . 9 (𝑁 ∈ β„• β†’ ((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) = -1)
3534neeq1d 3001 . . . . . . . 8 (𝑁 ∈ β„• β†’ (((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) β‰  0 ↔ -1 β‰  0))
3624, 35mpbiri 258 . . . . . . 7 (𝑁 ∈ β„• β†’ ((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) β‰  0)
37 ne0p 25721 . . . . . . 7 ((0 ∈ β„‚ ∧ ((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) β‰  0) β†’ (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) β‰  0𝑝)
3823, 36, 37sylancr 588 . . . . . 6 (𝑁 ∈ β„• β†’ (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) β‰  0𝑝)
3929mptiniseg 6239 . . . . . . . . 9 (0 ∈ β„‚ β†’ (β—‘(𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) β€œ {0}) = {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0})
4023, 39ax-mp 5 . . . . . . . 8 (β—‘(𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) β€œ {0}) = {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}
4140eqcomi 2742 . . . . . . 7 {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0} = (β—‘(𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) β€œ {0})
4241fta1 25821 . . . . . 6 (((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) ∈ (Polyβ€˜β„‚) ∧ (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) β‰  0𝑝) β†’ ({𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0} ∈ Fin ∧ (β™―β€˜{𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ≀ (degβ€˜(𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)))))
4322, 38, 42syl2anc 585 . . . . 5 (𝑁 ∈ β„• β†’ ({𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0} ∈ Fin ∧ (β™―β€˜{𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ≀ (degβ€˜(𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)))))
4443simpld 496 . . . 4 (𝑁 ∈ β„• β†’ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0} ∈ Fin)
45 unfi 9172 . . . 4 (({0} ∈ Fin ∧ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0} ∈ Fin) β†’ ({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ∈ Fin)
461, 44, 45sylancr 588 . . 3 (𝑁 ∈ β„• β†’ ({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ∈ Fin)
47 eqid 2733 . . . 4 (β„€/nβ„€β€˜π‘) = (β„€/nβ„€β€˜π‘)
48 eqid 2733 . . . 4 (Baseβ€˜(β„€/nβ„€β€˜π‘)) = (Baseβ€˜(β„€/nβ„€β€˜π‘))
4947, 48znfi 21115 . . 3 (𝑁 ∈ β„• β†’ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∈ Fin)
50 mapfi 9348 . . 3 ((({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ∈ Fin ∧ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∈ Fin) β†’ (({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ↑m (Baseβ€˜(β„€/nβ„€β€˜π‘))) ∈ Fin)
5146, 49, 50syl2anc 585 . 2 (𝑁 ∈ β„• β†’ (({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ↑m (Baseβ€˜(β„€/nβ„€β€˜π‘))) ∈ Fin)
52 dchrabl.g . . . . . . . 8 𝐺 = (DChrβ€˜π‘)
53 dchrfi.b . . . . . . . 8 𝐷 = (Baseβ€˜πΊ)
54 simpr 486 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ 𝑓 ∈ 𝐷)
5552, 47, 53, 48, 54dchrf 26745 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ 𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))βŸΆβ„‚)
5655ffnd 6719 . . . . . 6 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ 𝑓 Fn (Baseβ€˜(β„€/nβ„€β€˜π‘)))
57 df-ne 2942 . . . . . . . . . . 11 ((π‘“β€˜π‘₯) β‰  0 ↔ Β¬ (π‘“β€˜π‘₯) = 0)
58 fvex 6905 . . . . . . . . . . . 12 (π‘“β€˜π‘₯) ∈ V
5958elsn 4644 . . . . . . . . . . 11 ((π‘“β€˜π‘₯) ∈ {0} ↔ (π‘“β€˜π‘₯) = 0)
6057, 59xchbinxr 335 . . . . . . . . . 10 ((π‘“β€˜π‘₯) β‰  0 ↔ Β¬ (π‘“β€˜π‘₯) ∈ {0})
61 oveq1 7416 . . . . . . . . . . . . . 14 (𝑧 = (π‘“β€˜π‘₯) β†’ (𝑧↑(Ο•β€˜π‘)) = ((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)))
6261oveq1d 7424 . . . . . . . . . . . . 13 (𝑧 = (π‘“β€˜π‘₯) β†’ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = (((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)) βˆ’ 1))
6362eqeq1d 2735 . . . . . . . . . . . 12 (𝑧 = (π‘“β€˜π‘₯) β†’ (((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0 ↔ (((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)) βˆ’ 1) = 0))
64 simpl 484 . . . . . . . . . . . . 13 ((π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0) β†’ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)))
65 ffvelcdm 7084 . . . . . . . . . . . . 13 ((𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))βŸΆβ„‚ ∧ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))) β†’ (π‘“β€˜π‘₯) ∈ β„‚)
6655, 64, 65syl2an 597 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (π‘“β€˜π‘₯) ∈ β„‚)
6752, 47, 53dchrmhm 26744 . . . . . . . . . . . . . . . . . 18 𝐷 βŠ† ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) MndHom (mulGrpβ€˜β„‚fld))
68 simplr 768 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ 𝑓 ∈ 𝐷)
6967, 68sselid 3981 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ 𝑓 ∈ ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) MndHom (mulGrpβ€˜β„‚fld)))
7014ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (Ο•β€˜π‘) ∈ β„•0)
71 simprl 770 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)))
72 eqid 2733 . . . . . . . . . . . . . . . . . . 19 (mulGrpβ€˜(β„€/nβ„€β€˜π‘)) = (mulGrpβ€˜(β„€/nβ„€β€˜π‘))
7372, 48mgpbas 19993 . . . . . . . . . . . . . . . . . 18 (Baseβ€˜(β„€/nβ„€β€˜π‘)) = (Baseβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))
74 eqid 2733 . . . . . . . . . . . . . . . . . 18 (.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘))) = (.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))
75 eqid 2733 . . . . . . . . . . . . . . . . . 18 (.gβ€˜(mulGrpβ€˜β„‚fld)) = (.gβ€˜(mulGrpβ€˜β„‚fld))
7673, 74, 75mhmmulg 18995 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) MndHom (mulGrpβ€˜β„‚fld)) ∧ (Ο•β€˜π‘) ∈ β„•0 ∧ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))) β†’ (π‘“β€˜((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))π‘₯)) = ((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜β„‚fld))(π‘“β€˜π‘₯)))
7769, 70, 71, 76syl3anc 1372 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (π‘“β€˜((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))π‘₯)) = ((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜β„‚fld))(π‘“β€˜π‘₯)))
78 nnnn0 12479 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
7947zncrng 21100 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ β„•0 β†’ (β„€/nβ„€β€˜π‘) ∈ CRing)
8078, 79syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„• β†’ (β„€/nβ„€β€˜π‘) ∈ CRing)
81 crngring 20068 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„€/nβ„€β€˜π‘) ∈ CRing β†’ (β„€/nβ„€β€˜π‘) ∈ Ring)
8280, 81syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„• β†’ (β„€/nβ„€β€˜π‘) ∈ Ring)
8382ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (β„€/nβ„€β€˜π‘) ∈ Ring)
84 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . 23 (Unitβ€˜(β„€/nβ„€β€˜π‘)) = (Unitβ€˜(β„€/nβ„€β€˜π‘))
85 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . 23 ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))) = ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘)))
8684, 85unitgrp 20197 . . . . . . . . . . . . . . . . . . . . . 22 ((β„€/nβ„€β€˜π‘) ∈ Ring β†’ ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))) ∈ Grp)
8783, 86syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))) ∈ Grp)
8847, 84znunithash 21120 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„• β†’ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))) = (Ο•β€˜π‘))
8988, 14eqeltrd 2834 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„• β†’ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))) ∈ β„•0)
90 fvex 6905 . . . . . . . . . . . . . . . . . . . . . . . 24 (Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ V
91 hashclb 14318 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ V β†’ ((Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ Fin ↔ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))) ∈ β„•0))
9290, 91ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 ((Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ Fin ↔ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))) ∈ β„•0)
9389, 92sylibr 233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„• β†’ (Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ Fin)
9493ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ Fin)
95 simprr 772 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (π‘“β€˜π‘₯) β‰  0)
9652, 47, 53, 48, 84, 68, 71dchrn0 26753 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((π‘“β€˜π‘₯) β‰  0 ↔ π‘₯ ∈ (Unitβ€˜(β„€/nβ„€β€˜π‘))))
9795, 96mpbid 231 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ π‘₯ ∈ (Unitβ€˜(β„€/nβ„€β€˜π‘)))
9884, 85unitgrpbas 20196 . . . . . . . . . . . . . . . . . . . . . 22 (Unitβ€˜(β„€/nβ„€β€˜π‘)) = (Baseβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))
99 eqid 2733 . . . . . . . . . . . . . . . . . . . . . 22 (odβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘)))) = (odβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))
10098, 99oddvds2 19434 . . . . . . . . . . . . . . . . . . . . 21 ((((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))) ∈ Grp ∧ (Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ Fin ∧ π‘₯ ∈ (Unitβ€˜(β„€/nβ„€β€˜π‘))) β†’ ((odβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))β€˜π‘₯) βˆ₯ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))))
10187, 94, 97, 100syl3anc 1372 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((odβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))β€˜π‘₯) βˆ₯ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))))
10288ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))) = (Ο•β€˜π‘))
103101, 102breqtrd 5175 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((odβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))β€˜π‘₯) βˆ₯ (Ο•β€˜π‘))
10413ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (Ο•β€˜π‘) ∈ β„•)
105104nnzd 12585 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (Ο•β€˜π‘) ∈ β„€)
106 eqid 2733 . . . . . . . . . . . . . . . . . . . . 21 (.gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘)))) = (.gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))
107 eqid 2733 . . . . . . . . . . . . . . . . . . . . 21 (0gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘)))) = (0gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))
10898, 99, 106, 107oddvds 19415 . . . . . . . . . . . . . . . . . . . 20 ((((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))) ∈ Grp ∧ π‘₯ ∈ (Unitβ€˜(β„€/nβ„€β€˜π‘)) ∧ (Ο•β€˜π‘) ∈ β„€) β†’ (((odβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))β€˜π‘₯) βˆ₯ (Ο•β€˜π‘) ↔ ((Ο•β€˜π‘)(.gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))π‘₯) = (0gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))))
10987, 97, 105, 108syl3anc 1372 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (((odβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))β€˜π‘₯) βˆ₯ (Ο•β€˜π‘) ↔ ((Ο•β€˜π‘)(.gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))π‘₯) = (0gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))))
110103, 109mpbid 231 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((Ο•β€˜π‘)(.gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))π‘₯) = (0gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘)))))
11184, 72unitsubm 20200 . . . . . . . . . . . . . . . . . . . 20 ((β„€/nβ„€β€˜π‘) ∈ Ring β†’ (Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ (SubMndβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘))))
11283, 111syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ (SubMndβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘))))
11374, 85, 106submmulg 18998 . . . . . . . . . . . . . . . . . . 19 (((Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ (SubMndβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘))) ∧ (Ο•β€˜π‘) ∈ β„•0 ∧ π‘₯ ∈ (Unitβ€˜(β„€/nβ„€β€˜π‘))) β†’ ((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))π‘₯) = ((Ο•β€˜π‘)(.gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))π‘₯))
114112, 70, 97, 113syl3anc 1372 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))π‘₯) = ((Ο•β€˜π‘)(.gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))π‘₯))
115 eqid 2733 . . . . . . . . . . . . . . . . . . . . 21 (1rβ€˜(β„€/nβ„€β€˜π‘)) = (1rβ€˜(β„€/nβ„€β€˜π‘))
11672, 115ringidval 20006 . . . . . . . . . . . . . . . . . . . 20 (1rβ€˜(β„€/nβ„€β€˜π‘)) = (0gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))
11785, 116subm0 18696 . . . . . . . . . . . . . . . . . . 19 ((Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ (SubMndβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘))) β†’ (1rβ€˜(β„€/nβ„€β€˜π‘)) = (0gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘)))))
118112, 117syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (1rβ€˜(β„€/nβ„€β€˜π‘)) = (0gβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘)))))
119110, 114, 1183eqtr4d 2783 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))π‘₯) = (1rβ€˜(β„€/nβ„€β€˜π‘)))
120119fveq2d 6896 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (π‘“β€˜((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))π‘₯)) = (π‘“β€˜(1rβ€˜(β„€/nβ„€β€˜π‘))))
12177, 120eqtr3d 2775 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜β„‚fld))(π‘“β€˜π‘₯)) = (π‘“β€˜(1rβ€˜(β„€/nβ„€β€˜π‘))))
122 cnfldexp 20978 . . . . . . . . . . . . . . . 16 (((π‘“β€˜π‘₯) ∈ β„‚ ∧ (Ο•β€˜π‘) ∈ β„•0) β†’ ((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜β„‚fld))(π‘“β€˜π‘₯)) = ((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)))
12366, 70, 122syl2anc 585 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((Ο•β€˜π‘)(.gβ€˜(mulGrpβ€˜β„‚fld))(π‘“β€˜π‘₯)) = ((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)))
124 eqid 2733 . . . . . . . . . . . . . . . . . 18 (mulGrpβ€˜β„‚fld) = (mulGrpβ€˜β„‚fld)
125 cnfld1 20970 . . . . . . . . . . . . . . . . . 18 1 = (1rβ€˜β„‚fld)
126124, 125ringidval 20006 . . . . . . . . . . . . . . . . 17 1 = (0gβ€˜(mulGrpβ€˜β„‚fld))
127116, 126mhm0 18680 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) MndHom (mulGrpβ€˜β„‚fld)) β†’ (π‘“β€˜(1rβ€˜(β„€/nβ„€β€˜π‘))) = 1)
12869, 127syl 17 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (π‘“β€˜(1rβ€˜(β„€/nβ„€β€˜π‘))) = 1)
129121, 123, 1283eqtr3d 2781 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ ((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)) = 1)
130129oveq1d 7424 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)) βˆ’ 1) = (1 βˆ’ 1))
131 1m1e0 12284 . . . . . . . . . . . . 13 (1 βˆ’ 1) = 0
132130, 131eqtrdi 2789 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)) βˆ’ 1) = 0)
13363, 66, 132elrabd 3686 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (π‘“β€˜π‘₯) ∈ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0})
134133expr 458 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))) β†’ ((π‘“β€˜π‘₯) β‰  0 β†’ (π‘“β€˜π‘₯) ∈ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}))
13560, 134biimtrrid 242 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))) β†’ (Β¬ (π‘“β€˜π‘₯) ∈ {0} β†’ (π‘“β€˜π‘₯) ∈ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}))
136135orrd 862 . . . . . . . 8 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))) β†’ ((π‘“β€˜π‘₯) ∈ {0} ∨ (π‘“β€˜π‘₯) ∈ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}))
137 elun 4149 . . . . . . . 8 ((π‘“β€˜π‘₯) ∈ ({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ↔ ((π‘“β€˜π‘₯) ∈ {0} ∨ (π‘“β€˜π‘₯) ∈ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}))
138136, 137sylibr 233 . . . . . . 7 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))) β†’ (π‘“β€˜π‘₯) ∈ ({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}))
139138ralrimiva 3147 . . . . . 6 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ βˆ€π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))(π‘“β€˜π‘₯) ∈ ({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}))
140 ffnfv 7118 . . . . . 6 (𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))⟢({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ↔ (𝑓 Fn (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))(π‘“β€˜π‘₯) ∈ ({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0})))
14156, 139, 140sylanbrc 584 . . . . 5 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ 𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))⟢({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}))
142141ex 414 . . . 4 (𝑁 ∈ β„• β†’ (𝑓 ∈ 𝐷 β†’ 𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))⟢({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0})))
14346, 49elmapd 8834 . . . 4 (𝑁 ∈ β„• β†’ (𝑓 ∈ (({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ↑m (Baseβ€˜(β„€/nβ„€β€˜π‘))) ↔ 𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))⟢({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0})))
144142, 143sylibrd 259 . . 3 (𝑁 ∈ β„• β†’ (𝑓 ∈ 𝐷 β†’ 𝑓 ∈ (({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ↑m (Baseβ€˜(β„€/nβ„€β€˜π‘)))))
145144ssrdv 3989 . 2 (𝑁 ∈ β„• β†’ 𝐷 βŠ† (({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ↑m (Baseβ€˜(β„€/nβ„€β€˜π‘))))
14651, 145ssfid 9267 1 (𝑁 ∈ β„• β†’ 𝐷 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  {crab 3433  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  {csn 4629   class class class wbr 5149   ↦ cmpt 5232   Γ— cxp 5675  β—‘ccnv 5676   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∘f cof 7668   ↑m cmap 8820  Fincfn 8939  β„‚cc 11108  0cc0 11110  1c1 11111   ≀ cle 11249   βˆ’ cmin 11444  -cneg 11445  β„•cn 12212  β„•0cn0 12472  β„€cz 12558  β†‘cexp 14027  β™―chash 14290   βˆ₯ cdvds 16197  Ο•cphi 16697  Basecbs 17144   β†Ύs cress 17173  0gc0g 17385   MndHom cmhm 18669  SubMndcsubmnd 18670  Grpcgrp 18819  .gcmg 18950  odcod 19392  mulGrpcmgp 19987  1rcur 20004  Ringcrg 20056  CRingccrg 20057  Unitcui 20169  β„‚fldccnfld 20944  β„€/nβ„€czn 21052  0𝑝c0p 25186  Polycply 25698  degcdgr 25701  DChrcdchr 26735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-ec 8705  df-qs 8709  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-oi 9505  df-dju 9896  df-card 9934  df-acn 9937  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-xnn0 12545  df-z 12559  df-dec 12678  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-sum 15633  df-dvds 16198  df-gcd 16436  df-phi 16699  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-0g 17387  df-imas 17454  df-qus 17455  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-subg 19003  df-nsg 19004  df-eqg 19005  df-ghm 19090  df-od 19396  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-rnghom 20251  df-subrg 20317  df-lmod 20473  df-lss 20543  df-lsp 20583  df-sra 20785  df-rgmod 20786  df-lidl 20787  df-rsp 20788  df-2idl 20857  df-cnfld 20945  df-zring 21018  df-zrh 21053  df-zn 21056  df-0p 25187  df-ply 25702  df-idp 25703  df-coe 25704  df-dgr 25705  df-quot 25804  df-dchr 26736
This theorem is referenced by:  sumdchr2  26773  dchrhash  26774  rpvmasum2  27015  dchrisum0re  27016
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