Proof of Theorem dchrelbasd
Step | Hyp | Ref
| Expression |
1 | | dchrelbasd.5 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑋 ∈ ℂ) |
2 | 1 | adantlr 747 |
. . . 4
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → 𝑋 ∈ ℂ) |
3 | | 0cnd 9912 |
. . . 4
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → 0 ∈ ℂ) |
4 | 2, 3 | ifclda 4070 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, 𝑋, 0) ∈ ℂ) |
5 | | eqid 2610 |
. . 3
⊢ (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0)) = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0)) |
6 | 4, 5 | fmptd 6292 |
. 2
⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0)):𝐵⟶ℂ) |
7 | | dchrval.n |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℕ) |
8 | 7 | nnnn0d 11228 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
9 | | dchrval.z |
. . . . . . . . . 10
⊢ 𝑍 =
(ℤ/nℤ‘𝑁) |
10 | 9 | zncrng 19712 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ 𝑍 ∈
CRing) |
11 | | crngring 18381 |
. . . . . . . . 9
⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) |
12 | 8, 10, 11 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → 𝑍 ∈ Ring) |
13 | | dchrval.u |
. . . . . . . . . 10
⊢ 𝑈 = (Unit‘𝑍) |
14 | | eqid 2610 |
. . . . . . . . . 10
⊢
(.r‘𝑍) = (.r‘𝑍) |
15 | 13, 14 | unitmulcl 18487 |
. . . . . . . . 9
⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(.r‘𝑍)𝑦) ∈ 𝑈) |
16 | 15 | 3expb 1258 |
. . . . . . . 8
⊢ ((𝑍 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑥(.r‘𝑍)𝑦) ∈ 𝑈) |
17 | 12, 16 | sylan 487 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑥(.r‘𝑍)𝑦) ∈ 𝑈) |
18 | 17 | iftrued 4044 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → if((𝑥(.r‘𝑍)𝑦) ∈ 𝑈, 𝐸, 0) = 𝐸) |
19 | | dchrelbasd.6 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝐸 = (𝐴 · 𝐶)) |
20 | 18, 19 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → if((𝑥(.r‘𝑍)𝑦) ∈ 𝑈, 𝐸, 0) = (𝐴 · 𝐶)) |
21 | | dchrval.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝑍) |
22 | 21, 13 | unitss 18483 |
. . . . . . 7
⊢ 𝑈 ⊆ 𝐵 |
23 | 22, 17 | sseldi 3566 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑥(.r‘𝑍)𝑦) ∈ 𝐵) |
24 | 1 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ 𝑈 𝑋 ∈ ℂ) |
25 | 24 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∀𝑘 ∈ 𝑈 𝑋 ∈ ℂ) |
26 | | dchrelbasd.3 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑥(.r‘𝑍)𝑦) → 𝑋 = 𝐸) |
27 | 26 | eleq1d 2672 |
. . . . . . . . 9
⊢ (𝑘 = (𝑥(.r‘𝑍)𝑦) → (𝑋 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
28 | 27 | rspcv 3278 |
. . . . . . . 8
⊢ ((𝑥(.r‘𝑍)𝑦) ∈ 𝑈 → (∀𝑘 ∈ 𝑈 𝑋 ∈ ℂ → 𝐸 ∈ ℂ)) |
29 | 17, 25, 28 | sylc 63 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝐸 ∈ ℂ) |
30 | 18, 29 | eqeltrd 2688 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → if((𝑥(.r‘𝑍)𝑦) ∈ 𝑈, 𝐸, 0) ∈ ℂ) |
31 | | eleq1 2676 |
. . . . . . . 8
⊢ (𝑘 = (𝑥(.r‘𝑍)𝑦) → (𝑘 ∈ 𝑈 ↔ (𝑥(.r‘𝑍)𝑦) ∈ 𝑈)) |
32 | 31, 26 | ifbieq1d 4059 |
. . . . . . 7
⊢ (𝑘 = (𝑥(.r‘𝑍)𝑦) → if(𝑘 ∈ 𝑈, 𝑋, 0) = if((𝑥(.r‘𝑍)𝑦) ∈ 𝑈, 𝐸, 0)) |
33 | 32, 5 | fvmptg 6189 |
. . . . . 6
⊢ (((𝑥(.r‘𝑍)𝑦) ∈ 𝐵 ∧ if((𝑥(.r‘𝑍)𝑦) ∈ 𝑈, 𝐸, 0) ∈ ℂ) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘(𝑥(.r‘𝑍)𝑦)) = if((𝑥(.r‘𝑍)𝑦) ∈ 𝑈, 𝐸, 0)) |
34 | 23, 30, 33 | syl2anc 691 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘(𝑥(.r‘𝑍)𝑦)) = if((𝑥(.r‘𝑍)𝑦) ∈ 𝑈, 𝐸, 0)) |
35 | | simprl 790 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈) |
36 | 22, 35 | sseldi 3566 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝐵) |
37 | | iftrue 4042 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑈 → if(𝑥 ∈ 𝑈, 𝐴, 0) = 𝐴) |
38 | 37 | ad2antrl 760 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → if(𝑥 ∈ 𝑈, 𝐴, 0) = 𝐴) |
39 | | dchrelbasd.1 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑥 → 𝑋 = 𝐴) |
40 | 39 | eleq1d 2672 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑥 → (𝑋 ∈ ℂ ↔ 𝐴 ∈ ℂ)) |
41 | 40 | rspcv 3278 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑈 → (∀𝑘 ∈ 𝑈 𝑋 ∈ ℂ → 𝐴 ∈ ℂ)) |
42 | 35, 25, 41 | sylc 63 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝐴 ∈ ℂ) |
43 | 38, 42 | eqeltrd 2688 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → if(𝑥 ∈ 𝑈, 𝐴, 0) ∈ ℂ) |
44 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑈 ↔ 𝑥 ∈ 𝑈)) |
45 | 44, 39 | ifbieq1d 4059 |
. . . . . . . . 9
⊢ (𝑘 = 𝑥 → if(𝑘 ∈ 𝑈, 𝑋, 0) = if(𝑥 ∈ 𝑈, 𝐴, 0)) |
46 | 45, 5 | fvmptg 6189 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐵 ∧ if(𝑥 ∈ 𝑈, 𝐴, 0) ∈ ℂ) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑥) = if(𝑥 ∈ 𝑈, 𝐴, 0)) |
47 | 36, 43, 46 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑥) = if(𝑥 ∈ 𝑈, 𝐴, 0)) |
48 | 47, 38 | eqtrd 2644 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑥) = 𝐴) |
49 | | simprr 792 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈) |
50 | 22, 49 | sseldi 3566 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝐵) |
51 | | iftrue 4042 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝑈 → if(𝑦 ∈ 𝑈, 𝐶, 0) = 𝐶) |
52 | 51 | ad2antll 761 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → if(𝑦 ∈ 𝑈, 𝐶, 0) = 𝐶) |
53 | | dchrelbasd.2 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑦 → 𝑋 = 𝐶) |
54 | 53 | eleq1d 2672 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑦 → (𝑋 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
55 | 54 | rspcv 3278 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝑈 → (∀𝑘 ∈ 𝑈 𝑋 ∈ ℂ → 𝐶 ∈ ℂ)) |
56 | 49, 25, 55 | sylc 63 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝐶 ∈ ℂ) |
57 | 52, 56 | eqeltrd 2688 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → if(𝑦 ∈ 𝑈, 𝐶, 0) ∈ ℂ) |
58 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑦 → (𝑘 ∈ 𝑈 ↔ 𝑦 ∈ 𝑈)) |
59 | 58, 53 | ifbieq1d 4059 |
. . . . . . . . 9
⊢ (𝑘 = 𝑦 → if(𝑘 ∈ 𝑈, 𝑋, 0) = if(𝑦 ∈ 𝑈, 𝐶, 0)) |
60 | 59, 5 | fvmptg 6189 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝐵 ∧ if(𝑦 ∈ 𝑈, 𝐶, 0) ∈ ℂ) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑦) = if(𝑦 ∈ 𝑈, 𝐶, 0)) |
61 | 50, 57, 60 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑦) = if(𝑦 ∈ 𝑈, 𝐶, 0)) |
62 | 61, 52 | eqtrd 2644 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑦) = 𝐶) |
63 | 48, 62 | oveq12d 6567 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑥) · ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑦)) = (𝐴 · 𝐶)) |
64 | 20, 34, 63 | 3eqtr4d 2654 |
. . . 4
⊢ ((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))) |
65 | 64 | ralrimivva 2954 |
. . 3
⊢ (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))) |
66 | | eqid 2610 |
. . . . . . . 8
⊢
(1r‘𝑍) = (1r‘𝑍) |
67 | 13, 66 | 1unit 18481 |
. . . . . . 7
⊢ (𝑍 ∈ Ring →
(1r‘𝑍)
∈ 𝑈) |
68 | 12, 67 | syl 17 |
. . . . . 6
⊢ (𝜑 → (1r‘𝑍) ∈ 𝑈) |
69 | 22, 68 | sseldi 3566 |
. . . . 5
⊢ (𝜑 → (1r‘𝑍) ∈ 𝐵) |
70 | 68 | iftrued 4044 |
. . . . . . 7
⊢ (𝜑 →
if((1r‘𝑍)
∈ 𝑈, 𝑌, 0) = 𝑌) |
71 | | dchrelbasd.7 |
. . . . . . 7
⊢ (𝜑 → 𝑌 = 1) |
72 | 70, 71 | eqtrd 2644 |
. . . . . 6
⊢ (𝜑 →
if((1r‘𝑍)
∈ 𝑈, 𝑌, 0) = 1) |
73 | | ax-1cn 9873 |
. . . . . 6
⊢ 1 ∈
ℂ |
74 | 72, 73 | syl6eqel 2696 |
. . . . 5
⊢ (𝜑 →
if((1r‘𝑍)
∈ 𝑈, 𝑌, 0) ∈ ℂ) |
75 | | eleq1 2676 |
. . . . . . 7
⊢ (𝑘 = (1r‘𝑍) → (𝑘 ∈ 𝑈 ↔ (1r‘𝑍) ∈ 𝑈)) |
76 | | dchrelbasd.4 |
. . . . . . 7
⊢ (𝑘 = (1r‘𝑍) → 𝑋 = 𝑌) |
77 | 75, 76 | ifbieq1d 4059 |
. . . . . 6
⊢ (𝑘 = (1r‘𝑍) → if(𝑘 ∈ 𝑈, 𝑋, 0) = if((1r‘𝑍) ∈ 𝑈, 𝑌, 0)) |
78 | 77, 5 | fvmptg 6189 |
. . . . 5
⊢
(((1r‘𝑍) ∈ 𝐵 ∧ if((1r‘𝑍) ∈ 𝑈, 𝑌, 0) ∈ ℂ) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘(1r‘𝑍)) =
if((1r‘𝑍)
∈ 𝑈, 𝑌, 0)) |
79 | 69, 74, 78 | syl2anc 691 |
. . . 4
⊢ (𝜑 → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘(1r‘𝑍)) =
if((1r‘𝑍)
∈ 𝑈, 𝑌, 0)) |
80 | 79, 72 | eqtrd 2644 |
. . 3
⊢ (𝜑 → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘(1r‘𝑍)) = 1) |
81 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) |
82 | 24, 41 | mpan9 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ ℂ) |
83 | 82 | adantlr 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ ℂ) |
84 | | 0cnd 9912 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝑈) → 0 ∈ ℂ) |
85 | 83, 84 | ifclda 4070 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝑈, 𝐴, 0) ∈ ℂ) |
86 | 81, 85, 46 | syl2anc 691 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑥) = if(𝑥 ∈ 𝑈, 𝐴, 0)) |
87 | 86 | neeq1d 2841 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑥) ≠ 0 ↔ if(𝑥 ∈ 𝑈, 𝐴, 0) ≠ 0)) |
88 | | iffalse 4045 |
. . . . . 6
⊢ (¬
𝑥 ∈ 𝑈 → if(𝑥 ∈ 𝑈, 𝐴, 0) = 0) |
89 | 88 | necon1ai 2809 |
. . . . 5
⊢ (if(𝑥 ∈ 𝑈, 𝐴, 0) ≠ 0 → 𝑥 ∈ 𝑈) |
90 | 87, 89 | syl6bi 242 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)) |
91 | 90 | ralrimiva 2949 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (((𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0))‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)) |
92 | 65, 80, 91 | 3jca 1235 |
. 2
⊢ (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))) |
93 | | dchrval.g |
. . 3
⊢ 𝐺 = (DChr‘𝑁) |
94 | | dchrbas.b |
. . 3
⊢ 𝐷 = (Base‘𝐺) |
95 | 93, 9, 21, 13, 7, 94 | dchrelbas3 24763 |
. 2
⊢ (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))))) |
96 | 6, 92, 95 | mpbir2and 959 |
1
⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0)) ∈ 𝐷) |