Step |
Hyp |
Ref |
Expression |
1 |
|
dchrmhm.g |
⊢ 𝐺 = ( DChr ‘ 𝑁 ) |
2 |
|
dchrmhm.z |
⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 ) |
3 |
|
dchrmhm.b |
⊢ 𝐷 = ( Base ‘ 𝐺 ) |
4 |
|
dchrn0.b |
⊢ 𝐵 = ( Base ‘ 𝑍 ) |
5 |
|
dchrn0.u |
⊢ 𝑈 = ( Unit ‘ 𝑍 ) |
6 |
|
dchr1cl.o |
⊢ 1 = ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 1 , 0 ) ) |
7 |
|
dchrmulid2.t |
⊢ · = ( +g ‘ 𝐺 ) |
8 |
|
dchrmulid2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐷 ) |
9 |
|
dchrinvcl.n |
⊢ 𝐾 = ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) ) |
10 |
1 3
|
dchrrcl |
⊢ ( 𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ ) |
11 |
8 10
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
12 |
|
fveq2 |
⊢ ( 𝑘 = 𝑥 → ( 𝑋 ‘ 𝑘 ) = ( 𝑋 ‘ 𝑥 ) ) |
13 |
12
|
oveq2d |
⊢ ( 𝑘 = 𝑥 → ( 1 / ( 𝑋 ‘ 𝑘 ) ) = ( 1 / ( 𝑋 ‘ 𝑥 ) ) ) |
14 |
|
fveq2 |
⊢ ( 𝑘 = 𝑦 → ( 𝑋 ‘ 𝑘 ) = ( 𝑋 ‘ 𝑦 ) ) |
15 |
14
|
oveq2d |
⊢ ( 𝑘 = 𝑦 → ( 1 / ( 𝑋 ‘ 𝑘 ) ) = ( 1 / ( 𝑋 ‘ 𝑦 ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) → ( 𝑋 ‘ 𝑘 ) = ( 𝑋 ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) ) |
17 |
16
|
oveq2d |
⊢ ( 𝑘 = ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) → ( 1 / ( 𝑋 ‘ 𝑘 ) ) = ( 1 / ( 𝑋 ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) ) ) |
18 |
|
fveq2 |
⊢ ( 𝑘 = ( 1r ‘ 𝑍 ) → ( 𝑋 ‘ 𝑘 ) = ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) ) |
19 |
18
|
oveq2d |
⊢ ( 𝑘 = ( 1r ‘ 𝑍 ) → ( 1 / ( 𝑋 ‘ 𝑘 ) ) = ( 1 / ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) ) ) |
20 |
1 2 3 4 8
|
dchrf |
⊢ ( 𝜑 → 𝑋 : 𝐵 ⟶ ℂ ) |
21 |
4 5
|
unitss |
⊢ 𝑈 ⊆ 𝐵 |
22 |
21
|
sseli |
⊢ ( 𝑘 ∈ 𝑈 → 𝑘 ∈ 𝐵 ) |
23 |
|
ffvelrn |
⊢ ( ( 𝑋 : 𝐵 ⟶ ℂ ∧ 𝑘 ∈ 𝐵 ) → ( 𝑋 ‘ 𝑘 ) ∈ ℂ ) |
24 |
20 22 23
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → ( 𝑋 ‘ 𝑘 ) ∈ ℂ ) |
25 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → 𝑘 ∈ 𝑈 ) |
26 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → 𝑋 ∈ 𝐷 ) |
27 |
22
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → 𝑘 ∈ 𝐵 ) |
28 |
1 2 3 4 5 26 27
|
dchrn0 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → ( ( 𝑋 ‘ 𝑘 ) ≠ 0 ↔ 𝑘 ∈ 𝑈 ) ) |
29 |
25 28
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → ( 𝑋 ‘ 𝑘 ) ≠ 0 ) |
30 |
24 29
|
reccld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → ( 1 / ( 𝑋 ‘ 𝑘 ) ) ∈ ℂ ) |
31 |
|
1t1e1 |
⊢ ( 1 · 1 ) = 1 |
32 |
31
|
eqcomi |
⊢ 1 = ( 1 · 1 ) |
33 |
32
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 1 = ( 1 · 1 ) ) |
34 |
1 2 3
|
dchrmhm |
⊢ 𝐷 ⊆ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) |
35 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑋 ∈ 𝐷 ) |
36 |
34 35
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ) |
37 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑥 ∈ 𝑈 ) |
38 |
21 37
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑥 ∈ 𝐵 ) |
39 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑦 ∈ 𝑈 ) |
40 |
21 39
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑦 ∈ 𝐵 ) |
41 |
|
eqid |
⊢ ( mulGrp ‘ 𝑍 ) = ( mulGrp ‘ 𝑍 ) |
42 |
41 4
|
mgpbas |
⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑍 ) ) |
43 |
|
eqid |
⊢ ( .r ‘ 𝑍 ) = ( .r ‘ 𝑍 ) |
44 |
41 43
|
mgpplusg |
⊢ ( .r ‘ 𝑍 ) = ( +g ‘ ( mulGrp ‘ 𝑍 ) ) |
45 |
|
eqid |
⊢ ( mulGrp ‘ ℂfld ) = ( mulGrp ‘ ℂfld ) |
46 |
|
cnfldmul |
⊢ · = ( .r ‘ ℂfld ) |
47 |
45 46
|
mgpplusg |
⊢ · = ( +g ‘ ( mulGrp ‘ ℂfld ) ) |
48 |
42 44 47
|
mhmlin |
⊢ ( ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑋 ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = ( ( 𝑋 ‘ 𝑥 ) · ( 𝑋 ‘ 𝑦 ) ) ) |
49 |
36 38 40 48
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑋 ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = ( ( 𝑋 ‘ 𝑥 ) · ( 𝑋 ‘ 𝑦 ) ) ) |
50 |
33 49
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 1 / ( 𝑋 ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) ) = ( ( 1 · 1 ) / ( ( 𝑋 ‘ 𝑥 ) · ( 𝑋 ‘ 𝑦 ) ) ) ) |
51 |
|
1cnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 1 ∈ ℂ ) |
52 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑋 : 𝐵 ⟶ ℂ ) |
53 |
52 38
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑋 ‘ 𝑥 ) ∈ ℂ ) |
54 |
52 40
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑋 ‘ 𝑦 ) ∈ ℂ ) |
55 |
1 2 3 4 5 35 38
|
dchrn0 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 𝑋 ‘ 𝑥 ) ≠ 0 ↔ 𝑥 ∈ 𝑈 ) ) |
56 |
37 55
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑋 ‘ 𝑥 ) ≠ 0 ) |
57 |
1 2 3 4 5 35 40
|
dchrn0 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 𝑋 ‘ 𝑦 ) ≠ 0 ↔ 𝑦 ∈ 𝑈 ) ) |
58 |
39 57
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑋 ‘ 𝑦 ) ≠ 0 ) |
59 |
51 53 51 54 56 58
|
divmuldivd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 1 / ( 𝑋 ‘ 𝑥 ) ) · ( 1 / ( 𝑋 ‘ 𝑦 ) ) ) = ( ( 1 · 1 ) / ( ( 𝑋 ‘ 𝑥 ) · ( 𝑋 ‘ 𝑦 ) ) ) ) |
60 |
50 59
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 1 / ( 𝑋 ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) ) = ( ( 1 / ( 𝑋 ‘ 𝑥 ) ) · ( 1 / ( 𝑋 ‘ 𝑦 ) ) ) ) |
61 |
34 8
|
sselid |
⊢ ( 𝜑 → 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ) |
62 |
|
eqid |
⊢ ( 1r ‘ 𝑍 ) = ( 1r ‘ 𝑍 ) |
63 |
41 62
|
ringidval |
⊢ ( 1r ‘ 𝑍 ) = ( 0g ‘ ( mulGrp ‘ 𝑍 ) ) |
64 |
|
cnfld1 |
⊢ 1 = ( 1r ‘ ℂfld ) |
65 |
45 64
|
ringidval |
⊢ 1 = ( 0g ‘ ( mulGrp ‘ ℂfld ) ) |
66 |
63 65
|
mhm0 |
⊢ ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) → ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) = 1 ) |
67 |
61 66
|
syl |
⊢ ( 𝜑 → ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) = 1 ) |
68 |
67
|
oveq2d |
⊢ ( 𝜑 → ( 1 / ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) ) = ( 1 / 1 ) ) |
69 |
|
1div1e1 |
⊢ ( 1 / 1 ) = 1 |
70 |
68 69
|
eqtrdi |
⊢ ( 𝜑 → ( 1 / ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) ) = 1 ) |
71 |
1 2 4 5 11 3 13 15 17 19 30 60 70
|
dchrelbasd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) ) ∈ 𝐷 ) |
72 |
9 71
|
eqeltrid |
⊢ ( 𝜑 → 𝐾 ∈ 𝐷 ) |
73 |
1 2 3 7 72 8
|
dchrmul |
⊢ ( 𝜑 → ( 𝐾 · 𝑋 ) = ( 𝐾 ∘f · 𝑋 ) ) |
74 |
4
|
fvexi |
⊢ 𝐵 ∈ V |
75 |
74
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
76 |
|
ovex |
⊢ ( 1 / ( 𝑋 ‘ 𝑘 ) ) ∈ V |
77 |
|
c0ex |
⊢ 0 ∈ V |
78 |
76 77
|
ifex |
⊢ if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) ∈ V |
79 |
78
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) ∈ V ) |
80 |
20
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑋 ‘ 𝑘 ) ∈ ℂ ) |
81 |
9
|
a1i |
⊢ ( 𝜑 → 𝐾 = ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) ) ) |
82 |
20
|
feqmptd |
⊢ ( 𝜑 → 𝑋 = ( 𝑘 ∈ 𝐵 ↦ ( 𝑋 ‘ 𝑘 ) ) ) |
83 |
75 79 80 81 82
|
offval2 |
⊢ ( 𝜑 → ( 𝐾 ∘f · 𝑋 ) = ( 𝑘 ∈ 𝐵 ↦ ( if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) · ( 𝑋 ‘ 𝑘 ) ) ) ) |
84 |
|
ovif |
⊢ ( if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) · ( 𝑋 ‘ 𝑘 ) ) = if ( 𝑘 ∈ 𝑈 , ( ( 1 / ( 𝑋 ‘ 𝑘 ) ) · ( 𝑋 ‘ 𝑘 ) ) , ( 0 · ( 𝑋 ‘ 𝑘 ) ) ) |
85 |
80
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑘 ∈ 𝑈 ) → ( 𝑋 ‘ 𝑘 ) ∈ ℂ ) |
86 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝑋 ∈ 𝐷 ) |
87 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝑘 ∈ 𝐵 ) |
88 |
1 2 3 4 5 86 87
|
dchrn0 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( ( 𝑋 ‘ 𝑘 ) ≠ 0 ↔ 𝑘 ∈ 𝑈 ) ) |
89 |
88
|
biimpar |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑘 ∈ 𝑈 ) → ( 𝑋 ‘ 𝑘 ) ≠ 0 ) |
90 |
85 89
|
recid2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑘 ∈ 𝑈 ) → ( ( 1 / ( 𝑋 ‘ 𝑘 ) ) · ( 𝑋 ‘ 𝑘 ) ) = 1 ) |
91 |
90
|
ifeq1da |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝑈 , ( ( 1 / ( 𝑋 ‘ 𝑘 ) ) · ( 𝑋 ‘ 𝑘 ) ) , ( 0 · ( 𝑋 ‘ 𝑘 ) ) ) = if ( 𝑘 ∈ 𝑈 , 1 , ( 0 · ( 𝑋 ‘ 𝑘 ) ) ) ) |
92 |
80
|
mul02d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 0 · ( 𝑋 ‘ 𝑘 ) ) = 0 ) |
93 |
92
|
ifeq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝑈 , 1 , ( 0 · ( 𝑋 ‘ 𝑘 ) ) ) = if ( 𝑘 ∈ 𝑈 , 1 , 0 ) ) |
94 |
91 93
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝑈 , ( ( 1 / ( 𝑋 ‘ 𝑘 ) ) · ( 𝑋 ‘ 𝑘 ) ) , ( 0 · ( 𝑋 ‘ 𝑘 ) ) ) = if ( 𝑘 ∈ 𝑈 , 1 , 0 ) ) |
95 |
84 94
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) · ( 𝑋 ‘ 𝑘 ) ) = if ( 𝑘 ∈ 𝑈 , 1 , 0 ) ) |
96 |
95
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐵 ↦ ( if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) · ( 𝑋 ‘ 𝑘 ) ) ) = ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 1 , 0 ) ) ) |
97 |
6 96
|
eqtr4id |
⊢ ( 𝜑 → 1 = ( 𝑘 ∈ 𝐵 ↦ ( if ( 𝑘 ∈ 𝑈 , ( 1 / ( 𝑋 ‘ 𝑘 ) ) , 0 ) · ( 𝑋 ‘ 𝑘 ) ) ) ) |
98 |
83 97
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐾 ∘f · 𝑋 ) = 1 ) |
99 |
73 98
|
eqtrd |
⊢ ( 𝜑 → ( 𝐾 · 𝑋 ) = 1 ) |
100 |
72 99
|
jca |
⊢ ( 𝜑 → ( 𝐾 ∈ 𝐷 ∧ ( 𝐾 · 𝑋 ) = 1 ) ) |