Proof of Theorem ioorrnopnlem
| Step | Hyp | Ref
| Expression |
| 1 | | ioorrnopnlem.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ Fin) |
| 2 | | ioorrnopnlem.d |
. . . . 5
⊢ 𝐷 = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑘
∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) |
| 3 | 1, 2 | rrndsxmet 46318 |
. . . 4
⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ
↑m 𝑋))) |
| 4 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑖𝜑 |
| 5 | | reex 11246 |
. . . . . . 7
⊢ ℝ
∈ V |
| 6 | 5 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℝ ∈
V) |
| 7 | | ioossre 13448 |
. . . . . . 7
⊢ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ |
| 8 | 7 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ) |
| 9 | 4, 6, 8 | ixpssmapc 45078 |
. . . . 5
⊢ (𝜑 → X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ (ℝ ↑m 𝑋)) |
| 10 | | ioorrnopnlem.f |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 11 | 9, 10 | sseldd 3984 |
. . . 4
⊢ (𝜑 → 𝐹 ∈ (ℝ ↑m 𝑋)) |
| 12 | | ioorrnopnlem.e |
. . . . . 6
⊢ 𝐸 = inf(𝐻, ℝ, < ) |
| 13 | | ioorrnopnlem.h |
. . . . . . . . 9
⊢ 𝐻 = ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) |
| 14 | 13 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐻 = ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))) |
| 15 | | ioorrnopnlem.b |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
| 16 | 15 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ) |
| 17 | 10 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 18 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋) |
| 19 | | fvixp2 45204 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 20 | 17, 18, 19 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 21 | 7, 20 | sselid 3981 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ) |
| 22 | 16, 21 | resubcld 11691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ) |
| 23 | | ioorrnopnlem.a |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
| 24 | 23 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ) |
| 25 | 24 | rexrd 11311 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈
ℝ*) |
| 26 | 16 | rexrd 11311 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈
ℝ*) |
| 27 | | iooltub 45523 |
. . . . . . . . . . . . . 14
⊢ (((𝐴‘𝑖) ∈ ℝ* ∧ (𝐵‘𝑖) ∈ ℝ* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖)) |
| 28 | 25, 26, 20, 27 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖)) |
| 29 | 21, 16 | posdifd 11850 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) < (𝐵‘𝑖) ↔ 0 < ((𝐵‘𝑖) − (𝐹‘𝑖)))) |
| 30 | 28, 29 | mpbid 232 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐵‘𝑖) − (𝐹‘𝑖))) |
| 31 | 22, 30 | elrpd 13074 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈
ℝ+) |
| 32 | 21, 24 | resubcld 11691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ) |
| 33 | | ioogtlb 45508 |
. . . . . . . . . . . . . 14
⊢ (((𝐴‘𝑖) ∈ ℝ* ∧ (𝐵‘𝑖) ∈ ℝ* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖)) |
| 34 | 25, 26, 20, 33 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖)) |
| 35 | 24, 21 | posdifd 11850 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖) < (𝐹‘𝑖) ↔ 0 < ((𝐹‘𝑖) − (𝐴‘𝑖)))) |
| 36 | 34, 35 | mpbid 232 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐹‘𝑖) − (𝐴‘𝑖))) |
| 37 | 32, 36 | elrpd 13074 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈
ℝ+) |
| 38 | 31, 37 | ifcld 4572 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈
ℝ+) |
| 39 | 38 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑖 ∈ 𝑋 if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈
ℝ+) |
| 40 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) = (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) |
| 41 | 40 | rnmptss 7143 |
. . . . . . . . 9
⊢
(∀𝑖 ∈
𝑋 if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ℝ+ → ran
(𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ⊆
ℝ+) |
| 42 | 39, 41 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ⊆
ℝ+) |
| 43 | 14, 42 | eqsstrd 4018 |
. . . . . . 7
⊢ (𝜑 → 𝐻 ⊆
ℝ+) |
| 44 | | ltso 11341 |
. . . . . . . . 9
⊢ < Or
ℝ |
| 45 | 44 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → < Or
ℝ) |
| 46 | 40 | rnmptfi 45176 |
. . . . . . . . . 10
⊢ (𝑋 ∈ Fin → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ∈ Fin) |
| 47 | 1, 46 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ∈ Fin) |
| 48 | 13, 47 | eqeltrid 2845 |
. . . . . . . 8
⊢ (𝜑 → 𝐻 ∈ Fin) |
| 49 | | ioorrnopnlem.n |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ≠ ∅) |
| 50 | 4, 38, 40, 49 | rnmptn0 6264 |
. . . . . . . . 9
⊢ (𝜑 → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ≠ ∅) |
| 51 | 14, 50 | eqnetrd 3008 |
. . . . . . . 8
⊢ (𝜑 → 𝐻 ≠ ∅) |
| 52 | | rpssre 13042 |
. . . . . . . . . 10
⊢
ℝ+ ⊆ ℝ |
| 53 | 52 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℝ+
⊆ ℝ) |
| 54 | 43, 53 | sstrd 3994 |
. . . . . . . 8
⊢ (𝜑 → 𝐻 ⊆ ℝ) |
| 55 | | fiinfcl 9541 |
. . . . . . . 8
⊢ (( <
Or ℝ ∧ (𝐻 ∈
Fin ∧ 𝐻 ≠ ∅
∧ 𝐻 ⊆ ℝ))
→ inf(𝐻, ℝ, <
) ∈ 𝐻) |
| 56 | 45, 48, 51, 54, 55 | syl13anc 1374 |
. . . . . . 7
⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈ 𝐻) |
| 57 | 43, 56 | sseldd 3984 |
. . . . . 6
⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈
ℝ+) |
| 58 | 12, 57 | eqeltrid 2845 |
. . . . 5
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
| 59 | | rpxr 13044 |
. . . . 5
⊢ (𝐸 ∈ ℝ+
→ 𝐸 ∈
ℝ*) |
| 60 | 58, 59 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐸 ∈
ℝ*) |
| 61 | | eqid 2737 |
. . . . 5
⊢
(MetOpen‘𝐷) =
(MetOpen‘𝐷) |
| 62 | 61 | blopn 24513 |
. . . 4
⊢ ((𝐷 ∈
(∞Met‘(ℝ ↑m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑m 𝑋) ∧ 𝐸 ∈ ℝ*) → (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷)) |
| 63 | 3, 11, 60, 62 | syl3anc 1373 |
. . 3
⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷)) |
| 64 | | ioorrnopnlem.v |
. . . . 5
⊢ 𝑉 = (𝐹(ball‘𝐷)𝐸) |
| 65 | 64 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝑉 = (𝐹(ball‘𝐷)𝐸)) |
| 66 | 1 | rrxtopnfi 46302 |
. . . . 5
⊢ (𝜑 →
(TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑘
∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))) |
| 67 | 2 | eqcomi 2746 |
. . . . . . 7
⊢ (𝑓 ∈ (ℝ
↑m 𝑋),
𝑔 ∈ (ℝ
↑m 𝑋)
↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷 |
| 68 | 67 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑘
∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷) |
| 69 | 68 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ
↑m 𝑋),
𝑔 ∈ (ℝ
↑m 𝑋)
↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷)) |
| 70 | 66, 69 | eqtrd 2777 |
. . . 4
⊢ (𝜑 →
(TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘𝐷)) |
| 71 | 65, 70 | eleq12d 2835 |
. . 3
⊢ (𝜑 → (𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)) ↔ (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷))) |
| 72 | 63, 71 | mpbird 257 |
. 2
⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝑋))) |
| 73 | | xmetpsmet 24358 |
. . . . . 6
⊢ (𝐷 ∈
(∞Met‘(ℝ ↑m 𝑋)) → 𝐷 ∈ (PsMet‘(ℝ
↑m 𝑋))) |
| 74 | 3, 73 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐷 ∈ (PsMet‘(ℝ
↑m 𝑋))) |
| 75 | | blcntrps 24422 |
. . . . 5
⊢ ((𝐷 ∈ (PsMet‘(ℝ
↑m 𝑋))
∧ 𝐹 ∈ (ℝ
↑m 𝑋) ∧
𝐸 ∈
ℝ+) → 𝐹 ∈ (𝐹(ball‘𝐷)𝐸)) |
| 76 | 74, 11, 58, 75 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → 𝐹 ∈ (𝐹(ball‘𝐷)𝐸)) |
| 77 | 65 | eqcomd 2743 |
. . . 4
⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) = 𝑉) |
| 78 | 76, 77 | eleqtrd 2843 |
. . 3
⊢ (𝜑 → 𝐹 ∈ 𝑉) |
| 79 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑔𝜑 |
| 80 | | elmapfn 8905 |
. . . . . . . 8
⊢ (𝑔 ∈ (ℝ
↑m 𝑋)
→ 𝑔 Fn 𝑋) |
| 81 | 80 | 3ad2ant2 1135 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → 𝑔 Fn 𝑋) |
| 82 | 25 | 3ad2antl1 1186 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈
ℝ*) |
| 83 | 26 | 3ad2antl1 1186 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈
ℝ*) |
| 84 | | simpl2 1193 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝑔 ∈ (ℝ ↑m 𝑋)) |
| 85 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋) |
| 86 | | elmapi 8889 |
. . . . . . . . . . . 12
⊢ (𝑔 ∈ (ℝ
↑m 𝑋)
→ 𝑔:𝑋⟶ℝ) |
| 87 | 86 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑔 ∈ (ℝ
↑m 𝑋) ∧
𝑖 ∈ 𝑋) → 𝑔:𝑋⟶ℝ) |
| 88 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝑔 ∈ (ℝ
↑m 𝑋) ∧
𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋) |
| 89 | 87, 88 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝑔 ∈ (ℝ
↑m 𝑋) ∧
𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ) |
| 90 | 84, 85, 89 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ) |
| 91 | 24 | 3ad2antl1 1186 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ) |
| 92 | 52, 58 | sselid 3981 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ∈ ℝ) |
| 93 | 92 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ) |
| 94 | 21, 93 | resubcld 11691 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) ∈ ℝ) |
| 95 | 94 | 3ad2antl1 1186 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) ∈ ℝ) |
| 96 | 52, 38 | sselid 3981 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ℝ) |
| 97 | 12 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐸 = inf(𝐻, ℝ, < )) |
| 98 | | infxrrefi 45393 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐻 ⊆ ℝ ∧ 𝐻 ∈ Fin ∧ 𝐻 ≠ ∅) → inf(𝐻, ℝ*, < ) =
inf(𝐻, ℝ, <
)) |
| 99 | 54, 48, 51, 98 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → inf(𝐻, ℝ*, < ) = inf(𝐻, ℝ, <
)) |
| 100 | 99 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → inf(𝐻, ℝ, < ) = inf(𝐻, ℝ*, <
)) |
| 101 | 97, 100 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐸 = inf(𝐻, ℝ*, <
)) |
| 102 | 101 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 = inf(𝐻, ℝ*, <
)) |
| 103 | | ressxr 11305 |
. . . . . . . . . . . . . . . . . 18
⊢ ℝ
⊆ ℝ* |
| 104 | 103 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ℝ ⊆
ℝ*) |
| 105 | 54, 104 | sstrd 3994 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐻 ⊆
ℝ*) |
| 106 | 105 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐻 ⊆
ℝ*) |
| 107 | 38 | elexd 3504 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ V) |
| 108 | 40 | elrnmpt1 5971 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ 𝑋 ∧ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ V) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))) |
| 109 | 18, 107, 108 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))) |
| 110 | 109, 13 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ 𝐻) |
| 111 | | infxrlb 13376 |
. . . . . . . . . . . . . . 15
⊢ ((𝐻 ⊆ ℝ*
∧ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ 𝐻) → inf(𝐻, ℝ*, < ) ≤
if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) |
| 112 | 106, 110,
111 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → inf(𝐻, ℝ*, < ) ≤
if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) |
| 113 | 102, 112 | eqbrtrd 5165 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) |
| 114 | | min2 13232 |
. . . . . . . . . . . . . 14
⊢ ((((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ ∧ ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖))) |
| 115 | 22, 32, 114 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖))) |
| 116 | 93, 96, 32, 113, 115 | letrd 11418 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐹‘𝑖) − (𝐴‘𝑖))) |
| 117 | 93, 21, 24, 116 | lesubd 11867 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ ((𝐹‘𝑖) − 𝐸)) |
| 118 | 117 | 3ad2antl1 1186 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ ((𝐹‘𝑖) − 𝐸)) |
| 119 | 21 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ) |
| 120 | 89 | adantll 714 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ) |
| 121 | 119, 120 | resubcld 11691 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℝ) |
| 122 | 121 | 3adantl3 1169 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℝ) |
| 123 | 1, 2 | rrndsmet 46317 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ
↑m 𝑋))) |
| 124 | 123 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐷 ∈ (Met‘(ℝ
↑m 𝑋))) |
| 125 | 11 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ (ℝ ↑m 𝑋)) |
| 126 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑔 ∈ (ℝ ↑m 𝑋)) |
| 127 | | metcl 24342 |
. . . . . . . . . . . . . 14
⊢ ((𝐷 ∈ (Met‘(ℝ
↑m 𝑋))
∧ 𝐹 ∈ (ℝ
↑m 𝑋) ∧
𝑔 ∈ (ℝ
↑m 𝑋))
→ (𝐹𝐷𝑔) ∈ ℝ) |
| 128 | 124, 125,
126, 127 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) ∈ ℝ) |
| 129 | 128 | 3adantl3 1169 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) ∈ ℝ) |
| 130 | 93 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ) |
| 131 | 130 | 3adantl3 1169 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ) |
| 132 | 121 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℂ) |
| 133 | 132 | abscld 15475 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ∈ ℝ) |
| 134 | 121 | leabsd 15453 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖)))) |
| 135 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑋 ∈ Fin) |
| 136 | | ixpf 8960 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 ∈ X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝐹:𝑋⟶∪
𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 137 | 10, 136 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐹:𝑋⟶∪
𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 138 | 8 | ralrimiva 3146 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ) |
| 139 | | iunss 5045 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ ↔ ∀𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ) |
| 140 | 138, 139 | sylibr 234 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ) |
| 141 | 137, 140 | fssd 6753 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
| 142 | 141 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐹:𝑋⟶ℝ) |
| 143 | 126, 86 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑔:𝑋⟶ℝ) |
| 144 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋) |
| 145 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢
(dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋)) |
| 146 | 135, 142,
143, 144, 145 | rrnprjdstle 46316 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹(dist‘(ℝ^‘𝑋))𝑔)) |
| 147 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(ℝ^‘𝑋) =
(ℝ^‘𝑋) |
| 148 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℝ
↑m 𝑋) =
(ℝ ↑m 𝑋) |
| 149 | 147, 148 | rrxdsfi 25445 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑋 ∈ Fin →
(dist‘(ℝ^‘𝑋)) = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑘
∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
| 150 | 1, 149 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 →
(dist‘(ℝ^‘𝑋)) = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑘
∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
| 151 | 150, 68 | eqtrd 2777 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
(dist‘(ℝ^‘𝑋)) = 𝐷) |
| 152 | 151 | oveqd 7448 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐹(dist‘(ℝ^‘𝑋))𝑔) = (𝐹𝐷𝑔)) |
| 153 | 152 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹(dist‘(ℝ^‘𝑋))𝑔) = (𝐹𝐷𝑔)) |
| 154 | 146, 153 | breqtrd 5169 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹𝐷𝑔)) |
| 155 | 121, 133,
128, 134, 154 | letrd 11418 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (𝐹𝐷𝑔)) |
| 156 | 155 | 3adantl3 1169 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (𝐹𝐷𝑔)) |
| 157 | | simpl3 1194 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) < 𝐸) |
| 158 | 122, 129,
131, 156, 157 | lelttrd 11419 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸) |
| 159 | | ltsub23 11743 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑖) ∈ ℝ ∧ (𝑔‘𝑖) ∈ ℝ ∧ 𝐸 ∈ ℝ) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖))) |
| 160 | 119, 120,
130, 159 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖))) |
| 161 | 160 | 3adantl3 1169 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖))) |
| 162 | 158, 161 | mpbid 232 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖)) |
| 163 | 91, 95, 90, 118, 162 | lelttrd 11419 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝑔‘𝑖)) |
| 164 | 21, 93 | readdcld 11290 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ∈ ℝ) |
| 165 | 164 | 3ad2antl1 1186 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ∈ ℝ) |
| 166 | 16 | 3ad2antl1 1186 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ) |
| 167 | 120, 119 | resubcld 11691 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ∈ ℝ) |
| 168 | 167 | 3adantl3 1169 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ∈ ℝ) |
| 169 | 167 | leabsd 15453 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝑔‘𝑖) − (𝐹‘𝑖)))) |
| 170 | 120 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℂ) |
| 171 | 119 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℂ) |
| 172 | 170, 171 | abssubd 15492 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))) = (abs‘((𝐹‘𝑖) − (𝑔‘𝑖)))) |
| 173 | 169, 172 | breqtrd 5169 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖)))) |
| 174 | 167, 133,
128, 173, 154 | letrd 11418 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (𝐹𝐷𝑔)) |
| 175 | 174 | 3adantl3 1169 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (𝐹𝐷𝑔)) |
| 176 | 168, 129,
131, 175, 157 | lelttrd 11419 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) < 𝐸) |
| 177 | 119 | 3adantl3 1169 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ) |
| 178 | 90, 177, 131 | ltsubadd2d 11861 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (((𝑔‘𝑖) − (𝐹‘𝑖)) < 𝐸 ↔ (𝑔‘𝑖) < ((𝐹‘𝑖) + 𝐸))) |
| 179 | 176, 178 | mpbid 232 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < ((𝐹‘𝑖) + 𝐸)) |
| 180 | | min1 13231 |
. . . . . . . . . . . . . 14
⊢ ((((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ ∧ ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ≤ ((𝐵‘𝑖) − (𝐹‘𝑖))) |
| 181 | 22, 32, 180 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ≤ ((𝐵‘𝑖) − (𝐹‘𝑖))) |
| 182 | 93, 96, 22, 113, 181 | letrd 11418 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖))) |
| 183 | 21, 93, 16 | leaddsub2d 11865 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖) ↔ 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖)))) |
| 184 | 182, 183 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖)) |
| 185 | 184 | 3ad2antl1 1186 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖)) |
| 186 | 90, 165, 166, 179, 185 | ltletrd 11421 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < (𝐵‘𝑖)) |
| 187 | 82, 83, 90, 163, 186 | eliood 45511 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 188 | 187 | ralrimiva 3146 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 189 | 81, 188 | jca 511 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → (𝑔 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) |
| 190 | | vex 3484 |
. . . . . . 7
⊢ 𝑔 ∈ V |
| 191 | 190 | elixp 8944 |
. . . . . 6
⊢ (𝑔 ∈ X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ (𝑔 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) |
| 192 | 189, 191 | sylibr 234 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → 𝑔 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 193 | 79, 74, 11, 60, 192 | ballss3 45098 |
. . . 4
⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 194 | 65, 193 | eqsstrd 4018 |
. . 3
⊢ (𝜑 → 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) |
| 195 | 78, 194 | jca 511 |
. 2
⊢ (𝜑 → (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) |
| 196 | | eleq2 2830 |
. . . 4
⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉)) |
| 197 | | sseq1 4009 |
. . . 4
⊢ (𝑣 = 𝑉 → (𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) |
| 198 | 196, 197 | anbi12d 632 |
. . 3
⊢ (𝑣 = 𝑉 → ((𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) ↔ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))) |
| 199 | 198 | rspcev 3622 |
. 2
⊢ ((𝑉 ∈
(TopOpen‘(ℝ^‘𝑋)) ∧ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) |
| 200 | 72, 195, 199 | syl2anc 584 |
1
⊢ (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) |