Proof of Theorem ssblex
| Step | Hyp | Ref
| Expression |
|---|
| 1 | | simprl 535 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ 𝑅 ∈
ℝ+) |
| 2 | 1 | rphalfcld 10121 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ (𝑅 / 2) ∈
ℝ+) |
| 3 | | simprr 537 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ 𝑆 ∈
ℝ+) |
| 4 | | rpmincl 12021 |
. . 3
⊢ (((𝑅 / 2) ∈ ℝ+
∧ 𝑆 ∈
ℝ+) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈
ℝ+) |
| 5 | 2, 3, 4 | syl2anc 415 |
. 2
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈
ℝ+) |
| 6 | 5 | rpred 10108 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈
ℝ) |
| 7 | 2 | rpred 10108 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ (𝑅 / 2) ∈
ℝ) |
| 8 | 1 | rpred 10108 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ 𝑅 ∈
ℝ) |
| 9 | 3 | rpred 10108 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ 𝑆 ∈
ℝ) |
| 10 | | min1inf 12015 |
. . . 4
⊢ (((𝑅 / 2) ∈ ℝ ∧ 𝑆 ∈ ℝ) →
inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ (𝑅 / 2)) |
| 11 | 7, 9, 10 | syl2anc 415 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ (𝑅 / 2)) |
| 12 | 1 | rpgt0d 10111 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ 0 < 𝑅) |
| 13 | | halfpos 9541 |
. . . . 5
⊢ (𝑅 ∈ ℝ → (0 <
𝑅 ↔ (𝑅 / 2) < 𝑅)) |
| 14 | 8, 13 | syl 14 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ (0 < 𝑅 ↔
(𝑅 / 2) < 𝑅)) |
| 15 | 12, 14 | mpbid 147 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ (𝑅 / 2) < 𝑅) |
| 16 | 6, 7, 8, 11, 15 | lelttrd 8453 |
. 2
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅) |
| 17 | | simpl 109 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ (𝐷 ∈
(∞Met‘𝑋) ∧
𝑃 ∈ 𝑋)) |
| 18 | 5 | rpxrd 10109 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈
ℝ*) |
| 19 | 3 | rpxrd 10109 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ 𝑆 ∈
ℝ*) |
| 20 | | min2inf 12016 |
. . . 4
⊢ (((𝑅 / 2) ∈ ℝ ∧ 𝑆 ∈ ℝ) →
inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ 𝑆) |
| 21 | 7, 9, 20 | syl2anc 415 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ 𝑆) |
| 22 | | ssbl 15579 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈
ℝ* ∧ 𝑆
∈ ℝ*) ∧ inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ 𝑆) → (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆)) |
| 23 | 17, 18, 19, 21, 22 | syl121anc 1283 |
. 2
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆)) |
| 24 | | breq1 4133 |
. . . 4
⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑥 < 𝑅 ↔ inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅)) |
| 25 | | oveq2 6093 |
. . . . 5
⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < ))) |
| 26 | 25 | sseq1d 3277 |
. . . 4
⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆) ↔ (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))) |
| 27 | 24, 26 | anbi12d 477 |
. . 3
⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → ((𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)) ↔ (inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅 ∧ (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆)))) |
| 28 | 27 | rspcev 2929 |
. 2
⊢
((inf({(𝑅 / 2),
𝑆}, ℝ, < ) ∈
ℝ+ ∧ (inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅 ∧ (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆))) |
| 29 | 5, 16, 23, 28 | syl12anc 1276 |
1
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+))
→ ∃𝑥 ∈
ℝ+ (𝑥 <
𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆))) |
