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Theorem dchrfi 26982
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 9046 . . . 4 {0} ∈ Fin
2 cnex 11193 . . . . . . . . 9 β„‚ ∈ V
32a1i 11 . . . . . . . 8 (𝑁 ∈ β„• β†’ β„‚ ∈ V)
4 ovexd 7446 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑧 ∈ β„‚) β†’ (𝑧↑(Ο•β€˜π‘)) ∈ V)
5 1cnd 11213 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑧 ∈ β„‚) β†’ 1 ∈ β„‚)
6 eqidd 2733 . . . . . . . 8 (𝑁 ∈ β„• β†’ (𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) = (𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))))
7 fconstmpt 5738 . . . . . . . . 9 (β„‚ Γ— {1}) = (𝑧 ∈ β„‚ ↦ 1)
87a1i 11 . . . . . . . 8 (𝑁 ∈ β„• β†’ (β„‚ Γ— {1}) = (𝑧 ∈ β„‚ ↦ 1))
93, 4, 5, 6, 8offval2 7692 . . . . . . 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 4004 . . . . . . . . . 10 β„‚ βŠ† β„‚
1110a1i 11 . . . . . . . . 9 (𝑁 ∈ β„• β†’ β„‚ βŠ† β„‚)
12 1cnd 11213 . . . . . . . . 9 (𝑁 ∈ β„• β†’ 1 ∈ β„‚)
13 phicl 16706 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (Ο•β€˜π‘) ∈ β„•)
1413nnnn0d 12536 . . . . . . . . 9 (𝑁 ∈ β„• β†’ (Ο•β€˜π‘) ∈ β„•0)
15 plypow 25943 . . . . . . . . 9 ((β„‚ βŠ† β„‚ ∧ 1 ∈ β„‚ ∧ (Ο•β€˜π‘) ∈ β„•0) β†’ (𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∈ (Polyβ€˜β„‚))
1611, 12, 14, 15syl3anc 1371 . . . . . . . 8 (𝑁 ∈ β„• β†’ (𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∈ (Polyβ€˜β„‚))
17 ax-1cn 11170 . . . . . . . . 9 1 ∈ β„‚
18 plyconst 25944 . . . . . . . . 9 ((β„‚ βŠ† β„‚ ∧ 1 ∈ β„‚) β†’ (β„‚ Γ— {1}) ∈ (Polyβ€˜β„‚))
1910, 17, 18mp2an 690 . . . . . . . 8 (β„‚ Γ— {1}) ∈ (Polyβ€˜β„‚)
20 plysubcl 25960 . . . . . . . 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 586 . . . . . . 7 (𝑁 ∈ β„• β†’ ((𝑧 ∈ β„‚ ↦ (𝑧↑(Ο•β€˜π‘))) ∘f βˆ’ (β„‚ Γ— {1})) ∈ (Polyβ€˜β„‚))
229, 21eqeltrrd 2834 . . . . . 6 (𝑁 ∈ β„• β†’ (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) ∈ (Polyβ€˜β„‚))
23 0cn 11210 . . . . . . 7 0 ∈ β„‚
24 neg1ne0 12332 . . . . . . . 8 -1 β‰  0
25130expd 14108 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ (0↑(Ο•β€˜π‘)) = 0)
2625oveq1d 7426 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ ((0↑(Ο•β€˜π‘)) βˆ’ 1) = (0 βˆ’ 1))
27 oveq1 7418 . . . . . . . . . . . . 13 (𝑧 = 0 β†’ (𝑧↑(Ο•β€˜π‘)) = (0↑(Ο•β€˜π‘)))
2827oveq1d 7426 . . . . . . . . . . . 12 (𝑧 = 0 β†’ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = ((0↑(Ο•β€˜π‘)) βˆ’ 1))
29 eqid 2732 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) = (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))
30 ovex 7444 . . . . . . . . . . . 12 ((0↑(Ο•β€˜π‘)) βˆ’ 1) ∈ V
3128, 29, 30fvmpt 6998 . . . . . . . . . . 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 11451 . . . . . . . . . 10 -1 = (0 βˆ’ 1)
3426, 32, 333eqtr4g 2797 . . . . . . . . 9 (𝑁 ∈ β„• β†’ ((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) = -1)
3534neeq1d 3000 . . . . . . . 8 (𝑁 ∈ β„• β†’ (((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) β‰  0 ↔ -1 β‰  0))
3624, 35mpbiri 257 . . . . . . 7 (𝑁 ∈ β„• β†’ ((𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1))β€˜0) β‰  0)
37 ne0p 25945 . . . . . . 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 587 . . . . . 6 (𝑁 ∈ β„• β†’ (𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) β‰  0𝑝)
3929mptiniseg 6238 . . . . . . . . 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 2741 . . . . . . 7 {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0} = (β—‘(𝑧 ∈ β„‚ ↦ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1)) β€œ {0})
4241fta1 26045 . . . . . 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 584 . . . . 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 495 . . . 4 (𝑁 ∈ β„• β†’ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0} ∈ Fin)
45 unfi 9174 . . . 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 587 . . 3 (𝑁 ∈ β„• β†’ ({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}) ∈ Fin)
47 eqid 2732 . . . 4 (β„€/nβ„€β€˜π‘) = (β„€/nβ„€β€˜π‘)
48 eqid 2732 . . . 4 (Baseβ€˜(β„€/nβ„€β€˜π‘)) = (Baseβ€˜(β„€/nβ„€β€˜π‘))
4947, 48znfi 21334 . . 3 (𝑁 ∈ β„• β†’ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∈ Fin)
50 mapfi 9350 . . 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 584 . 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 485 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ 𝑓 ∈ 𝐷)
5552, 47, 53, 48, 54dchrf 26969 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ 𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))βŸΆβ„‚)
5655ffnd 6718 . . . . . 6 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ 𝑓 Fn (Baseβ€˜(β„€/nβ„€β€˜π‘)))
57 df-ne 2941 . . . . . . . . . . 11 ((π‘“β€˜π‘₯) β‰  0 ↔ Β¬ (π‘“β€˜π‘₯) = 0)
58 fvex 6904 . . . . . . . . . . . 12 (π‘“β€˜π‘₯) ∈ V
5958elsn 4643 . . . . . . . . . . 11 ((π‘“β€˜π‘₯) ∈ {0} ↔ (π‘“β€˜π‘₯) = 0)
6057, 59xchbinxr 334 . . . . . . . . . 10 ((π‘“β€˜π‘₯) β‰  0 ↔ Β¬ (π‘“β€˜π‘₯) ∈ {0})
61 oveq1 7418 . . . . . . . . . . . . . 14 (𝑧 = (π‘“β€˜π‘₯) β†’ (𝑧↑(Ο•β€˜π‘)) = ((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)))
6261oveq1d 7426 . . . . . . . . . . . . 13 (𝑧 = (π‘“β€˜π‘₯) β†’ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = (((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)) βˆ’ 1))
6362eqeq1d 2734 . . . . . . . . . . . 12 (𝑧 = (π‘“β€˜π‘₯) β†’ (((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0 ↔ (((π‘“β€˜π‘₯)↑(Ο•β€˜π‘)) βˆ’ 1) = 0))
64 simpl 483 . . . . . . . . . . . . 13 ((π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0) β†’ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)))
65 ffvelcdm 7083 . . . . . . . . . . . . 13 ((𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))βŸΆβ„‚ ∧ π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘))) β†’ (π‘“β€˜π‘₯) ∈ β„‚)
6655, 64, 65syl2an 596 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (π‘“β€˜π‘₯) ∈ β„‚)
6752, 47, 53dchrmhm 26968 . . . . . . . . . . . . . . . . . 18 𝐷 βŠ† ((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) MndHom (mulGrpβ€˜β„‚fld))
68 simplr 767 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ 𝑓 ∈ 𝐷)
6967, 68sselid 3980 . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (Ο•β€˜π‘) ∈ β„•0)
71 simprl 769 . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . 19 (mulGrpβ€˜(β„€/nβ„€β€˜π‘)) = (mulGrpβ€˜(β„€/nβ„€β€˜π‘))
7372, 48mgpbas 20034 . . . . . . . . . . . . . . . . . 18 (Baseβ€˜(β„€/nβ„€β€˜π‘)) = (Baseβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))
74 eqid 2732 . . . . . . . . . . . . . . . . . 18 (.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘))) = (.gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))
75 eqid 2732 . . . . . . . . . . . . . . . . . 18 (.gβ€˜(mulGrpβ€˜β„‚fld)) = (.gβ€˜(mulGrpβ€˜β„‚fld))
7673, 74, 75mhmmulg 19031 . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . 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 12483 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
7947zncrng 21319 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ β„•0 β†’ (β„€/nβ„€β€˜π‘) ∈ CRing)
8078, 79syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„• β†’ (β„€/nβ„€β€˜π‘) ∈ CRing)
81 crngring 20139 . . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (β„€/nβ„€β€˜π‘) ∈ Ring)
84 eqid 2732 . . . . . . . . . . . . . . . . . . . . . . 23 (Unitβ€˜(β„€/nβ„€β€˜π‘)) = (Unitβ€˜(β„€/nβ„€β€˜π‘))
85 eqid 2732 . . . . . . . . . . . . . . . . . . . . . . 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 20274 . . . . . . . . . . . . . . . . . . . . . 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 21339 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ β„• β†’ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))) = (Ο•β€˜π‘))
8988, 14eqeltrd 2833 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„• β†’ (β™―β€˜(Unitβ€˜(β„€/nβ„€β€˜π‘))) ∈ β„•0)
90 fvex 6904 . . . . . . . . . . . . . . . . . . . . . . . 24 (Unitβ€˜(β„€/nβ„€β€˜π‘)) ∈ V
91 hashclb 14322 . . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . . . . 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 26977 . . . . . . . . . . . . . . . . . . . . . 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 20273 . . . . . . . . . . . . . . . . . . . . . 22 (Unitβ€˜(β„€/nβ„€β€˜π‘)) = (Baseβ€˜((mulGrpβ€˜(β„€/nβ„€β€˜π‘)) β†Ύs (Unitβ€˜(β„€/nβ„€β€˜π‘))))
99 eqid 2732 . . . . . . . . . . . . . . . . . . . . . 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 19475 . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . 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 5174 . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (Ο•β€˜π‘) ∈ β„•)
105104nnzd 12589 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) ∧ (π‘₯ ∈ (Baseβ€˜(β„€/nβ„€β€˜π‘)) ∧ (π‘“β€˜π‘₯) β‰  0)) β†’ (Ο•β€˜π‘) ∈ β„€)
106 eqid 2732 . . . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . 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 19456 . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . 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 20277 . . . . . . . . . . . . . . . . . . . 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 19034 . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . 21 (1rβ€˜(β„€/nβ„€β€˜π‘)) = (1rβ€˜(β„€/nβ„€β€˜π‘))
11672, 115ringidval 20077 . . . . . . . . . . . . . . . . . . . 20 (1rβ€˜(β„€/nβ„€β€˜π‘)) = (0gβ€˜(mulGrpβ€˜(β„€/nβ„€β€˜π‘)))
11785, 116subm0 18732 . . . . . . . . . . . . . . . . . . 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 2782 . . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . 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 21178 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . 18 (mulGrpβ€˜β„‚fld) = (mulGrpβ€˜β„‚fld)
125 cnfld1 21170 . . . . . . . . . . . . . . . . . 18 1 = (1rβ€˜β„‚fld)
126124, 125ringidval 20077 . . . . . . . . . . . . . . . . 17 1 = (0gβ€˜(mulGrpβ€˜β„‚fld))
127116, 126mhm0 18716 . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . 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 7426 . . . . . . . . . . . . 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 12288 . . . . . . . . . . . . 13 (1 βˆ’ 1) = 0
132130, 131eqtrdi 2788 . . . . . . . . . . . 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 3685 . . . . . . . . . . 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 457 . . . . . . . . . 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 861 . . . . . . . 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 4148 . . . . . . . 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 3146 . . . . . 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 7120 . . . . . 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 583 . . . . 5 ((𝑁 ∈ β„• ∧ 𝑓 ∈ 𝐷) β†’ 𝑓:(Baseβ€˜(β„€/nβ„€β€˜π‘))⟢({0} βˆͺ {𝑧 ∈ β„‚ ∣ ((𝑧↑(Ο•β€˜π‘)) βˆ’ 1) = 0}))
142141ex 413 . . . 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 8836 . . . 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 258 . . 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 3988 . 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 9269 1 (𝑁 ∈ β„• β†’ 𝐷 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  {crab 3432  Vcvv 3474   βˆͺ cun 3946   βŠ† wss 3948  {csn 4628   class class class wbr 5148   ↦ cmpt 5231   Γ— cxp 5674  β—‘ccnv 5675   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∘f cof 7670   ↑m cmap 8822  Fincfn 8941  β„‚cc 11110  0cc0 11112  1c1 11113   ≀ cle 11253   βˆ’ cmin 11448  -cneg 11449  β„•cn 12216  β„•0cn0 12476  β„€cz 12562  β†‘cexp 14031  β™―chash 14294   βˆ₯ cdvds 16201  Ο•cphi 16701  Basecbs 17148   β†Ύs cress 17177  0gc0g 17389   MndHom cmhm 18703  SubMndcsubmnd 18704  Grpcgrp 18855  .gcmg 18986  odcod 19433  mulGrpcmgp 20028  1rcur 20075  Ringcrg 20127  CRingccrg 20128  Unitcui 20246  β„‚fldccnfld 21144  β„€/nβ„€czn 21271  0𝑝c0p 25410  Polycply 25922  degcdgr 25925  DChrcdchr 26959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  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  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-acn 9939  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-xnn0 12549  df-z 12563  df-dec 12682  df-uz 12827  df-rp 12979  df-fz 13489  df-fzo 13632  df-fl 13761  df-mod 13839  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-rlim 15437  df-sum 15637  df-dvds 16202  df-gcd 16440  df-phi 16703  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-0g 17391  df-imas 17458  df-qus 17459  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-mulg 18987  df-subg 19039  df-nsg 19040  df-eqg 19041  df-ghm 19128  df-od 19437  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-rhm 20363  df-subrng 20434  df-subrg 20459  df-lmod 20616  df-lss 20687  df-lsp 20727  df-sra 20930  df-rgmod 20931  df-lidl 20932  df-rsp 20933  df-2idl 21006  df-cnfld 21145  df-zring 21218  df-zrh 21272  df-zn 21275  df-0p 25411  df-ply 25926  df-idp 25927  df-coe 25928  df-dgr 25929  df-quot 26028  df-dchr 26960
This theorem is referenced by:  sumdchr2  26997  dchrhash  26998  rpvmasum2  27239  dchrisum0re  27240
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