Step | Hyp | Ref
| Expression |
1 | | dvconstbi.y |
. . . . . . 7
⊢ (𝜑 → 𝑌:𝑆⟶ℂ) |
2 | | dvconstbi.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
3 | | elpri 4197 |
. . . . . . . . 9
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 = ℝ ∨
𝑆 =
ℂ)) |
4 | 2, 3 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑆 = ℝ ∨ 𝑆 = ℂ)) |
5 | | 0re 10040 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
6 | | eleq2 2690 |
. . . . . . . . . 10
⊢ (𝑆 = ℝ → (0 ∈
𝑆 ↔ 0 ∈
ℝ)) |
7 | 5, 6 | mpbiri 248 |
. . . . . . . . 9
⊢ (𝑆 = ℝ → 0 ∈ 𝑆) |
8 | | 0cn 10032 |
. . . . . . . . . 10
⊢ 0 ∈
ℂ |
9 | | eleq2 2690 |
. . . . . . . . . 10
⊢ (𝑆 = ℂ → (0 ∈
𝑆 ↔ 0 ∈
ℂ)) |
10 | 8, 9 | mpbiri 248 |
. . . . . . . . 9
⊢ (𝑆 = ℂ → 0 ∈ 𝑆) |
11 | 7, 10 | jaoi 394 |
. . . . . . . 8
⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 0 ∈
𝑆) |
12 | 4, 11 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ 𝑆) |
13 | | ffvelrn 6357 |
. . . . . . 7
⊢ ((𝑌:𝑆⟶ℂ ∧ 0 ∈ 𝑆) → (𝑌‘0) ∈ ℂ) |
14 | 1, 12, 13 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → (𝑌‘0) ∈ ℂ) |
15 | 14 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (𝑌‘0) ∈ ℂ) |
16 | | ffn 6045 |
. . . . . . . 8
⊢ (𝑌:𝑆⟶ℂ → 𝑌 Fn 𝑆) |
17 | 1, 16 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑌 Fn 𝑆) |
18 | 17 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑌 Fn 𝑆) |
19 | | fvex 6201 |
. . . . . . 7
⊢ (𝑌‘0) ∈
V |
20 | | fnconstg 6093 |
. . . . . . 7
⊢ ((𝑌‘0) ∈ V → (𝑆 × {(𝑌‘0)}) Fn 𝑆) |
21 | 19, 20 | mp1i 13 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (𝑆 × {(𝑌‘0)}) Fn 𝑆) |
22 | 19 | fvconst2 6469 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝑆 → ((𝑆 × {(𝑌‘0)})‘𝑦) = (𝑌‘0)) |
23 | 22 | adantl 482 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ 𝑦 ∈ 𝑆) → ((𝑆 × {(𝑌‘0)})‘𝑦) = (𝑌‘0)) |
24 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((abs
∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) |
25 | 2, 24 | sblpnf 38509 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 0 ∈ 𝑆) → (0(ball‘((abs ∘ −
) ↾ (𝑆 × 𝑆)))+∞) = 𝑆) |
26 | 12, 25 | mpdan 702 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (0(ball‘((abs
∘ − ) ↾ (𝑆 × 𝑆)))+∞) = 𝑆) |
27 | 26 | eleq2d 2687 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞) ↔ 𝑦 ∈ 𝑆)) |
28 | 27 | biimpar 502 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) |
29 | 12, 26 | eleqtrrd 2704 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 ∈
(0(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))+∞)) |
30 | 2 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑆 ∈ {ℝ, ℂ}) |
31 | | ssid 3624 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑆 ⊆ 𝑆 |
32 | 31 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑆 ⊆ 𝑆) |
33 | 1 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑌:𝑆⟶ℂ) |
34 | 12 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 0 ∈ 𝑆) |
35 | | pnfxr 10092 |
. . . . . . . . . . . . . . . . . . 19
⊢ +∞
∈ ℝ* |
36 | 35 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → +∞ ∈
ℝ*) |
37 | | eqid 2622 |
. . . . . . . . . . . . . . . . . 18
⊢
(0(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))+∞) = (0(ball‘((abs ∘
− ) ↾ (𝑆
× 𝑆)))+∞) |
38 | 26 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (0(ball‘((abs
∘ − ) ↾ (𝑆 × 𝑆)))+∞) = 𝑆) |
39 | | dvconstbi.dy |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → dom (𝑆 D 𝑌) = 𝑆) |
40 | 39 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → dom (𝑆 D 𝑌) = 𝑆) |
41 | 38, 40 | eqtr4d 2659 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (0(ball‘((abs
∘ − ) ↾ (𝑆 × 𝑆)))+∞) = dom (𝑆 D 𝑌)) |
42 | | eqimss 3657 |
. . . . . . . . . . . . . . . . . . 19
⊢
((0(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))+∞) = dom (𝑆 D 𝑌) → (0(ball‘((abs ∘ −
) ↾ (𝑆 × 𝑆)))+∞) ⊆ dom (𝑆 D 𝑌)) |
43 | 41, 42 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (0(ball‘((abs
∘ − ) ↾ (𝑆 × 𝑆)))+∞) ⊆ dom (𝑆 D 𝑌)) |
44 | 5 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 0 ∈
ℝ) |
45 | 26 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞) ↔ 𝑥 ∈ 𝑆)) |
46 | 45 | biimpa 501 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) → 𝑥 ∈ 𝑆) |
47 | 46 | 3adant2 1080 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) → 𝑥 ∈ 𝑆) |
48 | | fveq1 6190 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑆 D 𝑌) = (𝑆 × {0}) → ((𝑆 D 𝑌)‘𝑥) = ((𝑆 × {0})‘𝑥)) |
49 | | c0ex 10034 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 0 ∈
V |
50 | 49 | fvconst2 6469 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ∈ 𝑆 → ((𝑆 × {0})‘𝑥) = 0) |
51 | 48, 50 | sylan9eq 2676 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → ((𝑆 D 𝑌)‘𝑥) = 0) |
52 | 51, 8 | syl6eqel 2709 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → ((𝑆 D 𝑌)‘𝑥) ∈ ℂ) |
53 | 52 | abscld 14175 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑆 D 𝑌)‘𝑥)) ∈ ℝ) |
54 | 51 | abs00bd 14031 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑆 D 𝑌)‘𝑥)) = 0) |
55 | | eqle 10139 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((abs‘((𝑆 D
𝑌)‘𝑥)) ∈ ℝ ∧ (abs‘((𝑆 D 𝑌)‘𝑥)) = 0) → (abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) |
56 | 53, 54, 55 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) |
57 | 56 | 3adant1 1079 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) |
58 | 47, 57 | syld3an3 1371 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) →
(abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) |
59 | 58 | 3expa 1265 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ 𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) →
(abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) |
60 | 30, 24, 32, 33, 34, 36, 37, 43, 44, 59 | dvlip2 23758 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ (0 ∈
(0(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))+∞) ∧ 𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞))) →
(abs‘((𝑌‘0)
− (𝑌‘𝑦))) ≤ (0 ·
(abs‘(0 − 𝑦)))) |
61 | 29, 60 | sylanr1 684 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ (𝜑 ∧ 𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞))) →
(abs‘((𝑌‘0)
− (𝑌‘𝑦))) ≤ (0 ·
(abs‘(0 − 𝑦)))) |
62 | 61 | 3impdi 1381 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) →
(abs‘((𝑌‘0)
− (𝑌‘𝑦))) ≤ (0 ·
(abs‘(0 − 𝑦)))) |
63 | 28, 62 | syl3an3 1361 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ (𝜑 ∧ 𝑦 ∈ 𝑆)) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ (0 · (abs‘(0 −
𝑦)))) |
64 | 63 | 3expa 1265 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ (𝜑 ∧ 𝑦 ∈ 𝑆)) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ (0 · (abs‘(0 −
𝑦)))) |
65 | 64 | 3impdi 1381 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ (0 · (abs‘(0 −
𝑦)))) |
66 | | recnprss 23668 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
67 | 2, 66 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
68 | 67 | sseld 3602 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℂ)) |
69 | | subcl 10280 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((0
∈ ℂ ∧ 𝑦
∈ ℂ) → (0 − 𝑦) ∈ ℂ) |
70 | 69 | abscld 14175 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℂ ∧ 𝑦
∈ ℂ) → (abs‘(0 − 𝑦)) ∈ ℝ) |
71 | 8, 70 | mpan 706 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ℂ →
(abs‘(0 − 𝑦))
∈ ℝ) |
72 | 68, 71 | syl6 35 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑦 ∈ 𝑆 → (abs‘(0 − 𝑦)) ∈
ℝ)) |
73 | 72 | imp 445 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(0 − 𝑦)) ∈
ℝ) |
74 | 73 | recnd 10068 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(0 − 𝑦)) ∈
ℂ) |
75 | 74 | mul02d 10234 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (0 · (abs‘(0 −
𝑦))) = 0) |
76 | 75 | 3adant2 1080 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (0 · (abs‘(0 −
𝑦))) = 0) |
77 | 65, 76 | breqtrd 4679 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ 0) |
78 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑌:𝑆⟶ℂ ∧ 𝑦 ∈ 𝑆) → (𝑌‘𝑦) ∈ ℂ) |
79 | 13, 78 | anim12dan 882 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑌:𝑆⟶ℂ ∧ (0 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) |
80 | 1, 79 | sylan 488 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (0 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) |
81 | 80 | 3impb 1260 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 0 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) |
82 | 12, 81 | syl3an2 1360 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) |
83 | 82 | 3anidm12 1383 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) |
84 | | subcl 10280 |
. . . . . . . . . . . . . 14
⊢ (((𝑌‘0) ∈ ℂ ∧
(𝑌‘𝑦) ∈ ℂ) → ((𝑌‘0) − (𝑌‘𝑦)) ∈ ℂ) |
85 | 83, 84 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) − (𝑌‘𝑦)) ∈ ℂ) |
86 | 85 | absge0d 14183 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 0 ≤ (abs‘((𝑌‘0) − (𝑌‘𝑦)))) |
87 | 86 | 3adant2 1080 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → 0 ≤ (abs‘((𝑌‘0) − (𝑌‘𝑦)))) |
88 | 85 | abscld 14175 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ∈ ℝ) |
89 | | letri3 10123 |
. . . . . . . . . . . . 13
⊢
(((abs‘((𝑌‘0) − (𝑌‘𝑦))) ∈ ℝ ∧ 0 ∈ ℝ)
→ ((abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0 ↔ ((abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ 0 ∧ 0 ≤ (abs‘((𝑌‘0) − (𝑌‘𝑦)))))) |
90 | 88, 5, 89 | sylancl 694 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0 ↔ ((abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ 0 ∧ 0 ≤ (abs‘((𝑌‘0) − (𝑌‘𝑦)))))) |
91 | 90 | 3adant2 1080 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → ((abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0 ↔ ((abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ 0 ∧ 0 ≤ (abs‘((𝑌‘0) − (𝑌‘𝑦)))))) |
92 | 77, 87, 91 | mpbir2and 957 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0) |
93 | 85 | abs00ad 14030 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0 ↔ ((𝑌‘0) − (𝑌‘𝑦)) = 0)) |
94 | 93 | 3adant2 1080 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → ((abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0 ↔ ((𝑌‘0) − (𝑌‘𝑦)) = 0)) |
95 | 92, 94 | mpbid 222 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) − (𝑌‘𝑦)) = 0) |
96 | | subeq0 10307 |
. . . . . . . . . . 11
⊢ (((𝑌‘0) ∈ ℂ ∧
(𝑌‘𝑦) ∈ ℂ) → (((𝑌‘0) − (𝑌‘𝑦)) = 0 ↔ (𝑌‘0) = (𝑌‘𝑦))) |
97 | 83, 96 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((𝑌‘0) − (𝑌‘𝑦)) = 0 ↔ (𝑌‘0) = (𝑌‘𝑦))) |
98 | 97 | 3adant2 1080 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (((𝑌‘0) − (𝑌‘𝑦)) = 0 ↔ (𝑌‘0) = (𝑌‘𝑦))) |
99 | 95, 98 | mpbid 222 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (𝑌‘0) = (𝑌‘𝑦)) |
100 | 99 | 3expa 1265 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ 𝑦 ∈ 𝑆) → (𝑌‘0) = (𝑌‘𝑦)) |
101 | 23, 100 | eqtr2d 2657 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ 𝑦 ∈ 𝑆) → (𝑌‘𝑦) = ((𝑆 × {(𝑌‘0)})‘𝑦)) |
102 | 18, 21, 101 | eqfnfvd 6314 |
. . . . 5
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑌 = (𝑆 × {(𝑌‘0)})) |
103 | | sneq 4187 |
. . . . . . . 8
⊢ (𝑥 = (𝑌‘0) → {𝑥} = {(𝑌‘0)}) |
104 | 103 | xpeq2d 5139 |
. . . . . . 7
⊢ (𝑥 = (𝑌‘0) → (𝑆 × {𝑥}) = (𝑆 × {(𝑌‘0)})) |
105 | 104 | eqeq2d 2632 |
. . . . . 6
⊢ (𝑥 = (𝑌‘0) → (𝑌 = (𝑆 × {𝑥}) ↔ 𝑌 = (𝑆 × {(𝑌‘0)}))) |
106 | 105 | rspcev 3309 |
. . . . 5
⊢ (((𝑌‘0) ∈ ℂ ∧
𝑌 = (𝑆 × {(𝑌‘0)})) → ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥})) |
107 | 15, 102, 106 | syl2anc 693 |
. . . 4
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥})) |
108 | 107 | ex 450 |
. . 3
⊢ (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) → ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥}))) |
109 | | oveq2 6658 |
. . . . . 6
⊢ (𝑌 = (𝑆 × {𝑥}) → (𝑆 D 𝑌) = (𝑆 D (𝑆 × {𝑥}))) |
110 | 109 | 3ad2ant3 1084 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑌 = (𝑆 × {𝑥})) → (𝑆 D 𝑌) = (𝑆 D (𝑆 × {𝑥}))) |
111 | | dvsconst 38529 |
. . . . . . 7
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧
𝑥 ∈ ℂ) →
(𝑆 D (𝑆 × {𝑥})) = (𝑆 × {0})) |
112 | 2, 111 | sylan 488 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑆 D (𝑆 × {𝑥})) = (𝑆 × {0})) |
113 | 112 | 3adant3 1081 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑌 = (𝑆 × {𝑥})) → (𝑆 D (𝑆 × {𝑥})) = (𝑆 × {0})) |
114 | 110, 113 | eqtrd 2656 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑌 = (𝑆 × {𝑥})) → (𝑆 D 𝑌) = (𝑆 × {0})) |
115 | 114 | rexlimdv3a 3033 |
. . 3
⊢ (𝜑 → (∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥}) → (𝑆 D 𝑌) = (𝑆 × {0}))) |
116 | 108, 115 | impbid 202 |
. 2
⊢ (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) ↔ ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥}))) |
117 | | sneq 4187 |
. . . . 5
⊢ (𝑐 = 𝑥 → {𝑐} = {𝑥}) |
118 | 117 | xpeq2d 5139 |
. . . 4
⊢ (𝑐 = 𝑥 → (𝑆 × {𝑐}) = (𝑆 × {𝑥})) |
119 | 118 | eqeq2d 2632 |
. . 3
⊢ (𝑐 = 𝑥 → (𝑌 = (𝑆 × {𝑐}) ↔ 𝑌 = (𝑆 × {𝑥}))) |
120 | 119 | cbvrexv 3172 |
. 2
⊢
(∃𝑐 ∈
ℂ 𝑌 = (𝑆 × {𝑐}) ↔ ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥})) |
121 | 116, 120 | syl6bbr 278 |
1
⊢ (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) ↔ ∃𝑐 ∈ ℂ 𝑌 = (𝑆 × {𝑐}))) |