| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | dvconstbi.y | . . . . . . 7
⊢ (𝜑 → 𝑌:𝑆⟶ℂ) | 
| 2 |  | dvconstbi.s | . . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | 
| 3 |  | elpri 4649 | . . . . . . . . 9
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 = ℝ ∨
𝑆 =
ℂ)) | 
| 4 | 2, 3 | syl 17 | . . . . . . . 8
⊢ (𝜑 → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | 
| 5 |  | 0re 11263 | . . . . . . . . . 10
⊢ 0 ∈
ℝ | 
| 6 |  | eleq2 2830 | . . . . . . . . . 10
⊢ (𝑆 = ℝ → (0 ∈
𝑆 ↔ 0 ∈
ℝ)) | 
| 7 | 5, 6 | mpbiri 258 | . . . . . . . . 9
⊢ (𝑆 = ℝ → 0 ∈ 𝑆) | 
| 8 |  | 0cn 11253 | . . . . . . . . . 10
⊢ 0 ∈
ℂ | 
| 9 |  | eleq2 2830 | . . . . . . . . . 10
⊢ (𝑆 = ℂ → (0 ∈
𝑆 ↔ 0 ∈
ℂ)) | 
| 10 | 8, 9 | mpbiri 258 | . . . . . . . . 9
⊢ (𝑆 = ℂ → 0 ∈ 𝑆) | 
| 11 | 7, 10 | jaoi 858 | . . . . . . . 8
⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 0 ∈
𝑆) | 
| 12 | 4, 11 | syl 17 | . . . . . . 7
⊢ (𝜑 → 0 ∈ 𝑆) | 
| 13 |  | ffvelcdm 7101 | . . . . . . 7
⊢ ((𝑌:𝑆⟶ℂ ∧ 0 ∈ 𝑆) → (𝑌‘0) ∈ ℂ) | 
| 14 | 1, 12, 13 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → (𝑌‘0) ∈ ℂ) | 
| 15 | 14 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (𝑌‘0) ∈ ℂ) | 
| 16 | 1 | ffnd 6737 | . . . . . . 7
⊢ (𝜑 → 𝑌 Fn 𝑆) | 
| 17 | 16 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑌 Fn 𝑆) | 
| 18 |  | fvex 6919 | . . . . . . 7
⊢ (𝑌‘0) ∈
V | 
| 19 |  | fnconstg 6796 | . . . . . . 7
⊢ ((𝑌‘0) ∈ V → (𝑆 × {(𝑌‘0)}) Fn 𝑆) | 
| 20 | 18, 19 | mp1i 13 | . . . . . 6
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (𝑆 × {(𝑌‘0)}) Fn 𝑆) | 
| 21 | 18 | fvconst2 7224 | . . . . . . . 8
⊢ (𝑦 ∈ 𝑆 → ((𝑆 × {(𝑌‘0)})‘𝑦) = (𝑌‘0)) | 
| 22 | 21 | adantl 481 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ 𝑦 ∈ 𝑆) → ((𝑆 × {(𝑌‘0)})‘𝑦) = (𝑌‘0)) | 
| 23 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . 19
⊢ ((abs
∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) | 
| 24 | 2, 23 | sblpnf 44329 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 0 ∈ 𝑆) → (0(ball‘((abs ∘ −
) ↾ (𝑆 × 𝑆)))+∞) = 𝑆) | 
| 25 | 12, 24 | mpdan 687 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (0(ball‘((abs
∘ − ) ↾ (𝑆 × 𝑆)))+∞) = 𝑆) | 
| 26 | 25 | eleq2d 2827 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞) ↔ 𝑦 ∈ 𝑆)) | 
| 27 | 26 | biimpar 477 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) | 
| 28 | 12, 25 | eleqtrrd 2844 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 ∈
(0(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))+∞)) | 
| 29 | 2 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑆 ∈ {ℝ, ℂ}) | 
| 30 |  | ssidd 4007 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑆 ⊆ 𝑆) | 
| 31 | 1 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑌:𝑆⟶ℂ) | 
| 32 | 12 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 0 ∈ 𝑆) | 
| 33 |  | pnfxr 11315 | . . . . . . . . . . . . . . . . . . 19
⊢ +∞
∈ ℝ* | 
| 34 | 33 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → +∞ ∈
ℝ*) | 
| 35 |  | eqid 2737 | . . . . . . . . . . . . . . . . . 18
⊢
(0(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))+∞) = (0(ball‘((abs ∘
− ) ↾ (𝑆
× 𝑆)))+∞) | 
| 36 | 25 | adantr 480 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (0(ball‘((abs
∘ − ) ↾ (𝑆 × 𝑆)))+∞) = 𝑆) | 
| 37 |  | dvconstbi.dy | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → dom (𝑆 D 𝑌) = 𝑆) | 
| 38 | 37 | adantr 480 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → dom (𝑆 D 𝑌) = 𝑆) | 
| 39 | 36, 38 | eqtr4d 2780 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (0(ball‘((abs
∘ − ) ↾ (𝑆 × 𝑆)))+∞) = dom (𝑆 D 𝑌)) | 
| 40 |  | eqimss 4042 | . . . . . . . . . . . . . . . . . . 19
⊢
((0(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))+∞) = dom (𝑆 D 𝑌) → (0(ball‘((abs ∘ −
) ↾ (𝑆 × 𝑆)))+∞) ⊆ dom (𝑆 D 𝑌)) | 
| 41 | 39, 40 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → (0(ball‘((abs
∘ − ) ↾ (𝑆 × 𝑆)))+∞) ⊆ dom (𝑆 D 𝑌)) | 
| 42 | 5 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 0 ∈
ℝ) | 
| 43 | 25 | eleq2d 2827 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞) ↔ 𝑥 ∈ 𝑆)) | 
| 44 | 43 | biimpa 476 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) → 𝑥 ∈ 𝑆) | 
| 45 | 44 | 3adant2 1132 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) → 𝑥 ∈ 𝑆) | 
| 46 |  | fveq1 6905 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑆 D 𝑌) = (𝑆 × {0}) → ((𝑆 D 𝑌)‘𝑥) = ((𝑆 × {0})‘𝑥)) | 
| 47 |  | c0ex 11255 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 0 ∈
V | 
| 48 | 47 | fvconst2 7224 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ∈ 𝑆 → ((𝑆 × {0})‘𝑥) = 0) | 
| 49 | 46, 48 | sylan9eq 2797 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → ((𝑆 D 𝑌)‘𝑥) = 0) | 
| 50 | 49, 8 | eqeltrdi 2849 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → ((𝑆 D 𝑌)‘𝑥) ∈ ℂ) | 
| 51 | 50 | abscld 15475 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑆 D 𝑌)‘𝑥)) ∈ ℝ) | 
| 52 | 49 | abs00bd 15330 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑆 D 𝑌)‘𝑥)) = 0) | 
| 53 |  | eqle 11363 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((abs‘((𝑆 D
𝑌)‘𝑥)) ∈ ℝ ∧ (abs‘((𝑆 D 𝑌)‘𝑥)) = 0) → (abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) | 
| 54 | 51, 52, 53 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) | 
| 55 | 54 | 3adant1 1131 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) | 
| 56 | 45, 55 | syld3an3 1411 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) →
(abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) | 
| 57 | 56 | 3expa 1119 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ 𝑥 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) →
(abs‘((𝑆 D 𝑌)‘𝑥)) ≤ 0) | 
| 58 | 29, 23, 30, 31, 32, 34, 35, 41, 42, 57 | dvlip2 26034 | . . . . . . . . . . . . . . . . 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)))) | 
| 59 | 28, 58 | sylanr1 682 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ (𝜑 ∧ 𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞))) →
(abs‘((𝑌‘0)
− (𝑌‘𝑦))) ≤ (0 ·
(abs‘(0 − 𝑦)))) | 
| 60 | 59 | 3impdi 1351 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ (0(ball‘((abs ∘ − )
↾ (𝑆 × 𝑆)))+∞)) →
(abs‘((𝑌‘0)
− (𝑌‘𝑦))) ≤ (0 ·
(abs‘(0 − 𝑦)))) | 
| 61 | 27, 60 | syl3an3 1166 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ (𝜑 ∧ 𝑦 ∈ 𝑆)) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ (0 · (abs‘(0 −
𝑦)))) | 
| 62 | 61 | 3expa 1119 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ (𝜑 ∧ 𝑦 ∈ 𝑆)) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ (0 · (abs‘(0 −
𝑦)))) | 
| 63 | 62 | 3impdi 1351 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ (0 · (abs‘(0 −
𝑦)))) | 
| 64 |  | recnprss 25939 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) | 
| 65 | 2, 64 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑆 ⊆ ℂ) | 
| 66 | 65 | sseld 3982 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℂ)) | 
| 67 |  | subcl 11507 | . . . . . . . . . . . . . . . . . . 19
⊢ ((0
∈ ℂ ∧ 𝑦
∈ ℂ) → (0 − 𝑦) ∈ ℂ) | 
| 68 | 67 | abscld 15475 | . . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℂ ∧ 𝑦
∈ ℂ) → (abs‘(0 − 𝑦)) ∈ ℝ) | 
| 69 | 8, 68 | mpan 690 | . . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ℂ →
(abs‘(0 − 𝑦))
∈ ℝ) | 
| 70 | 66, 69 | syl6 35 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑦 ∈ 𝑆 → (abs‘(0 − 𝑦)) ∈
ℝ)) | 
| 71 | 70 | imp 406 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(0 − 𝑦)) ∈
ℝ) | 
| 72 | 71 | recnd 11289 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(0 − 𝑦)) ∈
ℂ) | 
| 73 | 72 | mul02d 11459 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (0 · (abs‘(0 −
𝑦))) = 0) | 
| 74 | 73 | 3adant2 1132 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (0 · (abs‘(0 −
𝑦))) = 0) | 
| 75 | 63, 74 | breqtrd 5169 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ 0) | 
| 76 |  | ffvelcdm 7101 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑌:𝑆⟶ℂ ∧ 𝑦 ∈ 𝑆) → (𝑌‘𝑦) ∈ ℂ) | 
| 77 | 13, 76 | anim12dan 619 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑌:𝑆⟶ℂ ∧ (0 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) | 
| 78 | 1, 77 | sylan 580 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (0 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) | 
| 79 | 78 | 3impb 1115 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 0 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) | 
| 80 | 12, 79 | syl3an2 1165 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) | 
| 81 | 80 | 3anidm12 1421 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) ∈ ℂ ∧ (𝑌‘𝑦) ∈ ℂ)) | 
| 82 |  | subcl 11507 | . . . . . . . . . . . . . 14
⊢ (((𝑌‘0) ∈ ℂ ∧
(𝑌‘𝑦) ∈ ℂ) → ((𝑌‘0) − (𝑌‘𝑦)) ∈ ℂ) | 
| 83 | 81, 82 | syl 17 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) − (𝑌‘𝑦)) ∈ ℂ) | 
| 84 | 83 | absge0d 15483 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 0 ≤ (abs‘((𝑌‘0) − (𝑌‘𝑦)))) | 
| 85 | 84 | 3adant2 1132 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → 0 ≤ (abs‘((𝑌‘0) − (𝑌‘𝑦)))) | 
| 86 | 83 | abscld 15475 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) ∈ ℝ) | 
| 87 |  | letri3 11346 | . . . . . . . . . . . . 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)))))) | 
| 88 | 86, 5, 87 | sylancl 586 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0 ↔ ((abs‘((𝑌‘0) − (𝑌‘𝑦))) ≤ 0 ∧ 0 ≤ (abs‘((𝑌‘0) − (𝑌‘𝑦)))))) | 
| 89 | 88 | 3adant2 1132 | . . . . . . . . . . 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)))))) | 
| 90 | 75, 85, 89 | mpbir2and 713 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0) | 
| 91 | 83 | abs00ad 15329 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0 ↔ ((𝑌‘0) − (𝑌‘𝑦)) = 0)) | 
| 92 | 91 | 3adant2 1132 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → ((abs‘((𝑌‘0) − (𝑌‘𝑦))) = 0 ↔ ((𝑌‘0) − (𝑌‘𝑦)) = 0)) | 
| 93 | 90, 92 | mpbid 232 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → ((𝑌‘0) − (𝑌‘𝑦)) = 0) | 
| 94 |  | subeq0 11535 | . . . . . . . . . . 11
⊢ (((𝑌‘0) ∈ ℂ ∧
(𝑌‘𝑦) ∈ ℂ) → (((𝑌‘0) − (𝑌‘𝑦)) = 0 ↔ (𝑌‘0) = (𝑌‘𝑦))) | 
| 95 | 81, 94 | syl 17 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((𝑌‘0) − (𝑌‘𝑦)) = 0 ↔ (𝑌‘0) = (𝑌‘𝑦))) | 
| 96 | 95 | 3adant2 1132 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (((𝑌‘0) − (𝑌‘𝑦)) = 0 ↔ (𝑌‘0) = (𝑌‘𝑦))) | 
| 97 | 93, 96 | mpbid 232 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0}) ∧ 𝑦 ∈ 𝑆) → (𝑌‘0) = (𝑌‘𝑦)) | 
| 98 | 97 | 3expa 1119 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ 𝑦 ∈ 𝑆) → (𝑌‘0) = (𝑌‘𝑦)) | 
| 99 | 22, 98 | eqtr2d 2778 | . . . . . 6
⊢ (((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) ∧ 𝑦 ∈ 𝑆) → (𝑌‘𝑦) = ((𝑆 × {(𝑌‘0)})‘𝑦)) | 
| 100 | 17, 20, 99 | eqfnfvd 7054 | . . . . 5
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → 𝑌 = (𝑆 × {(𝑌‘0)})) | 
| 101 |  | sneq 4636 | . . . . . . 7
⊢ (𝑥 = (𝑌‘0) → {𝑥} = {(𝑌‘0)}) | 
| 102 | 101 | xpeq2d 5715 | . . . . . 6
⊢ (𝑥 = (𝑌‘0) → (𝑆 × {𝑥}) = (𝑆 × {(𝑌‘0)})) | 
| 103 | 102 | rspceeqv 3645 | . . . . 5
⊢ (((𝑌‘0) ∈ ℂ ∧
𝑌 = (𝑆 × {(𝑌‘0)})) → ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥})) | 
| 104 | 15, 100, 103 | syl2anc 584 | . . . 4
⊢ ((𝜑 ∧ (𝑆 D 𝑌) = (𝑆 × {0})) → ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥})) | 
| 105 | 104 | ex 412 | . . 3
⊢ (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) → ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥}))) | 
| 106 |  | oveq2 7439 | . . . . . 6
⊢ (𝑌 = (𝑆 × {𝑥}) → (𝑆 D 𝑌) = (𝑆 D (𝑆 × {𝑥}))) | 
| 107 | 106 | 3ad2ant3 1136 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑌 = (𝑆 × {𝑥})) → (𝑆 D 𝑌) = (𝑆 D (𝑆 × {𝑥}))) | 
| 108 |  | dvsconst 44349 | . . . . . . 7
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧
𝑥 ∈ ℂ) →
(𝑆 D (𝑆 × {𝑥})) = (𝑆 × {0})) | 
| 109 | 2, 108 | sylan 580 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑆 D (𝑆 × {𝑥})) = (𝑆 × {0})) | 
| 110 | 109 | 3adant3 1133 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑌 = (𝑆 × {𝑥})) → (𝑆 D (𝑆 × {𝑥})) = (𝑆 × {0})) | 
| 111 | 107, 110 | eqtrd 2777 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑌 = (𝑆 × {𝑥})) → (𝑆 D 𝑌) = (𝑆 × {0})) | 
| 112 | 111 | rexlimdv3a 3159 | . . 3
⊢ (𝜑 → (∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥}) → (𝑆 D 𝑌) = (𝑆 × {0}))) | 
| 113 | 105, 112 | impbid 212 | . 2
⊢ (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) ↔ ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥}))) | 
| 114 |  | sneq 4636 | . . . . 5
⊢ (𝑐 = 𝑥 → {𝑐} = {𝑥}) | 
| 115 | 114 | xpeq2d 5715 | . . . 4
⊢ (𝑐 = 𝑥 → (𝑆 × {𝑐}) = (𝑆 × {𝑥})) | 
| 116 | 115 | eqeq2d 2748 | . . 3
⊢ (𝑐 = 𝑥 → (𝑌 = (𝑆 × {𝑐}) ↔ 𝑌 = (𝑆 × {𝑥}))) | 
| 117 | 116 | cbvrexvw 3238 | . 2
⊢
(∃𝑐 ∈
ℂ 𝑌 = (𝑆 × {𝑐}) ↔ ∃𝑥 ∈ ℂ 𝑌 = (𝑆 × {𝑥})) | 
| 118 | 113, 117 | bitr4di 289 | 1
⊢ (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) ↔ ∃𝑐 ∈ ℂ 𝑌 = (𝑆 × {𝑐}))) |