Step |
Hyp |
Ref |
Expression |
1 |
|
dchrval.g |
⊢ 𝐺 = ( DChr ‘ 𝑁 ) |
2 |
|
dchrval.z |
⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 ) |
3 |
|
dchrval.b |
⊢ 𝐵 = ( Base ‘ 𝑍 ) |
4 |
|
dchrval.u |
⊢ 𝑈 = ( Unit ‘ 𝑍 ) |
5 |
|
dchrval.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
6 |
|
dchrbas.b |
⊢ 𝐷 = ( Base ‘ 𝐺 ) |
7 |
|
dchrelbasd.1 |
⊢ ( 𝑘 = 𝑥 → 𝑋 = 𝐴 ) |
8 |
|
dchrelbasd.2 |
⊢ ( 𝑘 = 𝑦 → 𝑋 = 𝐶 ) |
9 |
|
dchrelbasd.3 |
⊢ ( 𝑘 = ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) → 𝑋 = 𝐸 ) |
10 |
|
dchrelbasd.4 |
⊢ ( 𝑘 = ( 1r ‘ 𝑍 ) → 𝑋 = 𝑌 ) |
11 |
|
dchrelbasd.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → 𝑋 ∈ ℂ ) |
12 |
|
dchrelbasd.6 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝐸 = ( 𝐴 · 𝐶 ) ) |
13 |
|
dchrelbasd.7 |
⊢ ( 𝜑 → 𝑌 = 1 ) |
14 |
11
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑘 ∈ 𝑈 ) → 𝑋 ∈ ℂ ) |
15 |
|
0cnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ ¬ 𝑘 ∈ 𝑈 ) → 0 ∈ ℂ ) |
16 |
14 15
|
ifclda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ∈ ℂ ) |
17 |
16
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) : 𝐵 ⟶ ℂ ) |
18 |
5
|
nnnn0d |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
19 |
2
|
zncrng |
⊢ ( 𝑁 ∈ ℕ0 → 𝑍 ∈ CRing ) |
20 |
|
crngring |
⊢ ( 𝑍 ∈ CRing → 𝑍 ∈ Ring ) |
21 |
18 19 20
|
3syl |
⊢ ( 𝜑 → 𝑍 ∈ Ring ) |
22 |
|
eqid |
⊢ ( .r ‘ 𝑍 ) = ( .r ‘ 𝑍 ) |
23 |
4 22
|
unitmulcl |
⊢ ( ( 𝑍 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) → ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 ) |
24 |
23
|
3expb |
⊢ ( ( 𝑍 ∈ Ring ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 ) |
25 |
21 24
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 ) |
26 |
25
|
iftrued |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → if ( ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 , 𝐸 , 0 ) = 𝐸 ) |
27 |
26 12
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → if ( ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 , 𝐸 , 0 ) = ( 𝐴 · 𝐶 ) ) |
28 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) = ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) |
29 |
|
eleq1 |
⊢ ( 𝑘 = ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) → ( 𝑘 ∈ 𝑈 ↔ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 ) ) |
30 |
29 9
|
ifbieq1d |
⊢ ( 𝑘 = ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) → if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) = if ( ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 , 𝐸 , 0 ) ) |
31 |
3 4
|
unitss |
⊢ 𝑈 ⊆ 𝐵 |
32 |
31 25
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝐵 ) |
33 |
9
|
eleq1d |
⊢ ( 𝑘 = ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) → ( 𝑋 ∈ ℂ ↔ 𝐸 ∈ ℂ ) ) |
34 |
11
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝑈 𝑋 ∈ ℂ ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ∀ 𝑘 ∈ 𝑈 𝑋 ∈ ℂ ) |
36 |
33 35 25
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝐸 ∈ ℂ ) |
37 |
26 36
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → if ( ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 , 𝐸 , 0 ) ∈ ℂ ) |
38 |
28 30 32 37
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = if ( ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ∈ 𝑈 , 𝐸 , 0 ) ) |
39 |
|
eleq1 |
⊢ ( 𝑘 = 𝑥 → ( 𝑘 ∈ 𝑈 ↔ 𝑥 ∈ 𝑈 ) ) |
40 |
39 7
|
ifbieq1d |
⊢ ( 𝑘 = 𝑥 → if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) = if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) ) |
41 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑥 ∈ 𝑈 ) |
42 |
31 41
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑥 ∈ 𝐵 ) |
43 |
|
iftrue |
⊢ ( 𝑥 ∈ 𝑈 → if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) = 𝐴 ) |
44 |
43
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) = 𝐴 ) |
45 |
7
|
eleq1d |
⊢ ( 𝑘 = 𝑥 → ( 𝑋 ∈ ℂ ↔ 𝐴 ∈ ℂ ) ) |
46 |
45 35 41
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝐴 ∈ ℂ ) |
47 |
44 46
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) ∈ ℂ ) |
48 |
28 40 42 47
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) = if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) ) |
49 |
48 44
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) = 𝐴 ) |
50 |
|
eleq1 |
⊢ ( 𝑘 = 𝑦 → ( 𝑘 ∈ 𝑈 ↔ 𝑦 ∈ 𝑈 ) ) |
51 |
50 8
|
ifbieq1d |
⊢ ( 𝑘 = 𝑦 → if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) = if ( 𝑦 ∈ 𝑈 , 𝐶 , 0 ) ) |
52 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑦 ∈ 𝑈 ) |
53 |
31 52
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝑦 ∈ 𝐵 ) |
54 |
|
iftrue |
⊢ ( 𝑦 ∈ 𝑈 → if ( 𝑦 ∈ 𝑈 , 𝐶 , 0 ) = 𝐶 ) |
55 |
54
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → if ( 𝑦 ∈ 𝑈 , 𝐶 , 0 ) = 𝐶 ) |
56 |
8
|
eleq1d |
⊢ ( 𝑘 = 𝑦 → ( 𝑋 ∈ ℂ ↔ 𝐶 ∈ ℂ ) ) |
57 |
56 35 52
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → 𝐶 ∈ ℂ ) |
58 |
55 57
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → if ( 𝑦 ∈ 𝑈 , 𝐶 , 0 ) ∈ ℂ ) |
59 |
28 51 53 58
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑦 ) = if ( 𝑦 ∈ 𝑈 , 𝐶 , 0 ) ) |
60 |
59 55
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑦 ) = 𝐶 ) |
61 |
49 60
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) · ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑦 ) ) = ( 𝐴 · 𝐶 ) ) |
62 |
27 38 61
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) ) → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) · ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑦 ) ) ) |
63 |
62
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) · ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑦 ) ) ) |
64 |
|
eleq1 |
⊢ ( 𝑘 = ( 1r ‘ 𝑍 ) → ( 𝑘 ∈ 𝑈 ↔ ( 1r ‘ 𝑍 ) ∈ 𝑈 ) ) |
65 |
64 10
|
ifbieq1d |
⊢ ( 𝑘 = ( 1r ‘ 𝑍 ) → if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) = if ( ( 1r ‘ 𝑍 ) ∈ 𝑈 , 𝑌 , 0 ) ) |
66 |
|
eqid |
⊢ ( 1r ‘ 𝑍 ) = ( 1r ‘ 𝑍 ) |
67 |
4 66
|
1unit |
⊢ ( 𝑍 ∈ Ring → ( 1r ‘ 𝑍 ) ∈ 𝑈 ) |
68 |
21 67
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑍 ) ∈ 𝑈 ) |
69 |
31 68
|
sselid |
⊢ ( 𝜑 → ( 1r ‘ 𝑍 ) ∈ 𝐵 ) |
70 |
68
|
iftrued |
⊢ ( 𝜑 → if ( ( 1r ‘ 𝑍 ) ∈ 𝑈 , 𝑌 , 0 ) = 𝑌 ) |
71 |
70 13
|
eqtrd |
⊢ ( 𝜑 → if ( ( 1r ‘ 𝑍 ) ∈ 𝑈 , 𝑌 , 0 ) = 1 ) |
72 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
73 |
71 72
|
eqeltrdi |
⊢ ( 𝜑 → if ( ( 1r ‘ 𝑍 ) ∈ 𝑈 , 𝑌 , 0 ) ∈ ℂ ) |
74 |
28 65 69 73
|
fvmptd3 |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 1r ‘ 𝑍 ) ) = if ( ( 1r ‘ 𝑍 ) ∈ 𝑈 , 𝑌 , 0 ) ) |
75 |
74 71
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 1r ‘ 𝑍 ) ) = 1 ) |
76 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
77 |
45
|
rspcv |
⊢ ( 𝑥 ∈ 𝑈 → ( ∀ 𝑘 ∈ 𝑈 𝑋 ∈ ℂ → 𝐴 ∈ ℂ ) ) |
78 |
34 77
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → 𝐴 ∈ ℂ ) |
79 |
78
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑈 ) → 𝐴 ∈ ℂ ) |
80 |
|
0cnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ¬ 𝑥 ∈ 𝑈 ) → 0 ∈ ℂ ) |
81 |
79 80
|
ifclda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) ∈ ℂ ) |
82 |
28 40 76 81
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) = if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) ) |
83 |
82
|
neeq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) ≠ 0 ↔ if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) ≠ 0 ) ) |
84 |
|
iffalse |
⊢ ( ¬ 𝑥 ∈ 𝑈 → if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) = 0 ) |
85 |
84
|
necon1ai |
⊢ ( if ( 𝑥 ∈ 𝑈 , 𝐴 , 0 ) ≠ 0 → 𝑥 ∈ 𝑈 ) |
86 |
83 85
|
syl6bi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ 𝑈 ) ) |
87 |
86
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ 𝑈 ) ) |
88 |
63 75 87
|
3jca |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) · ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑦 ) ) ∧ ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 1r ‘ 𝑍 ) ) = 1 ∧ ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ 𝑈 ) ) ) |
89 |
1 2 3 4 5 6
|
dchrelbas3 |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ∈ 𝐷 ↔ ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) : 𝐵 ⟶ ℂ ∧ ( ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) · ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑦 ) ) ∧ ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ ( 1r ‘ 𝑍 ) ) = 1 ∧ ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ 𝑈 ) ) ) ) ) |
90 |
17 88 89
|
mpbir2and |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐵 ↦ if ( 𝑘 ∈ 𝑈 , 𝑋 , 0 ) ) ∈ 𝐷 ) |