| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | p0ex 5383 | . . . . . 6
⊢ {∅}
∈ V | 
| 2 | 1 | prid2 4762 | . . . . 5
⊢ {∅}
∈ {∅, {∅}} | 
| 3 | 2 | a1i 11 | . . . 4
⊢ (𝑋 = ∅ → {∅}
∈ {∅, {∅}}) | 
| 4 |  | ixpeq1 8949 | . . . . . 6
⊢ (𝑋 = ∅ → X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) | 
| 5 |  | ixp0x 8967 | . . . . . . 7
⊢ X𝑖 ∈
∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅} | 
| 6 | 5 | a1i 11 | . . . . . 6
⊢ (𝑋 = ∅ → X𝑖 ∈
∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅}) | 
| 7 | 4, 6 | eqtrd 2776 | . . . . 5
⊢ (𝑋 = ∅ → X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅}) | 
| 8 |  | 2fveq3 6910 | . . . . . 6
⊢ (𝑋 = ∅ →
(TopOpen‘(ℝ^‘𝑋)) =
(TopOpen‘(ℝ^‘∅))) | 
| 9 |  | rrxtopn0b 46316 | . . . . . . 7
⊢
(TopOpen‘(ℝ^‘∅)) = {∅,
{∅}} | 
| 10 | 9 | a1i 11 | . . . . . 6
⊢ (𝑋 = ∅ →
(TopOpen‘(ℝ^‘∅)) = {∅,
{∅}}) | 
| 11 | 8, 10 | eqtrd 2776 | . . . . 5
⊢ (𝑋 = ∅ →
(TopOpen‘(ℝ^‘𝑋)) = {∅, {∅}}) | 
| 12 | 7, 11 | eleq12d 2834 | . . . 4
⊢ (𝑋 = ∅ → (X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈
(TopOpen‘(ℝ^‘𝑋)) ↔ {∅} ∈ {∅,
{∅}})) | 
| 13 | 3, 12 | mpbird 257 | . . 3
⊢ (𝑋 = ∅ → X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈
(TopOpen‘(ℝ^‘𝑋))) | 
| 14 | 13 | adantl 481 | . 2
⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈
(TopOpen‘(ℝ^‘𝑋))) | 
| 15 |  | neqne 2947 | . . . 4
⊢ (¬
𝑋 = ∅ → 𝑋 ≠ ∅) | 
| 16 | 15 | adantl 481 | . . 3
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) | 
| 17 |  | fveq2 6905 | . . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → (𝐴‘𝑖) = (𝐴‘𝑗)) | 
| 18 |  | fveq2 6905 | . . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗)) | 
| 19 | 17, 18 | oveq12d 7450 | . . . . . . . . . 10
⊢ (𝑖 = 𝑗 → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = ((𝐴‘𝑗)(,)(𝐵‘𝑗))) | 
| 20 | 19 | cbvixpv 8956 | . . . . . . . . 9
⊢ X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗)) | 
| 21 | 20 | eleq2i 2832 | . . . . . . . 8
⊢ (𝑓 ∈ X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) | 
| 22 | 21 | biimpi 216 | . . . . . . 7
⊢ (𝑓 ∈ X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) | 
| 23 | 22 | adantl 481 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) | 
| 24 |  | ioorrnopn.x | . . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ Fin) | 
| 25 | 24 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → 𝑋 ∈ Fin) | 
| 26 | 21, 25 | sylan2br 595 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝑋 ∈ Fin) | 
| 27 |  | simplr 768 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → 𝑋 ≠ ∅) | 
| 28 | 21, 27 | sylan2br 595 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝑋 ≠ ∅) | 
| 29 |  | ioorrnopn.a | . . . . . . . . 9
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | 
| 30 | 29 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → 𝐴:𝑋⟶ℝ) | 
| 31 | 21, 30 | sylan2br 595 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝐴:𝑋⟶ℝ) | 
| 32 |  | ioorrnopn.b | . . . . . . . . 9
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | 
| 33 | 32 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → 𝐵:𝑋⟶ℝ) | 
| 34 | 21, 33 | sylan2br 595 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝐵:𝑋⟶ℝ) | 
| 35 |  | simpr 484 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) | 
| 36 | 21, 35 | sylan2br 595 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) | 
| 37 |  | eqid 2736 | . . . . . . 7
⊢ ran
(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)))) | 
| 38 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑗 = 𝑖 → (𝐵‘𝑗) = (𝐵‘𝑖)) | 
| 39 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑗 = 𝑖 → (𝑓‘𝑗) = (𝑓‘𝑖)) | 
| 40 | 38, 39 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑖 → ((𝐵‘𝑗) − (𝑓‘𝑗)) = ((𝐵‘𝑖) − (𝑓‘𝑖))) | 
| 41 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑗 = 𝑖 → (𝐴‘𝑗) = (𝐴‘𝑖)) | 
| 42 | 39, 41 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑖 → ((𝑓‘𝑗) − (𝐴‘𝑗)) = ((𝑓‘𝑖) − (𝐴‘𝑖))) | 
| 43 | 40, 42 | breq12d 5155 | . . . . . . . . . . 11
⊢ (𝑗 = 𝑖 → (((𝐵‘𝑗) − (𝑓‘𝑗)) ≤ ((𝑓‘𝑗) − (𝐴‘𝑗)) ↔ ((𝐵‘𝑖) − (𝑓‘𝑖)) ≤ ((𝑓‘𝑖) − (𝐴‘𝑖)))) | 
| 44 | 43, 40, 42 | ifbieq12d 4553 | . . . . . . . . . 10
⊢ (𝑗 = 𝑖 → if(((𝐵‘𝑗) − (𝑓‘𝑗)) ≤ ((𝑓‘𝑗) − (𝐴‘𝑗)), ((𝐵‘𝑗) − (𝑓‘𝑗)), ((𝑓‘𝑗) − (𝐴‘𝑗))) = if(((𝐵‘𝑖) − (𝑓‘𝑖)) ≤ ((𝑓‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝑓‘𝑖)), ((𝑓‘𝑖) − (𝐴‘𝑖)))) | 
| 45 | 44 | cbvmptv 5254 | . . . . . . . . 9
⊢ (𝑗 ∈ 𝑋 ↦ if(((𝐵‘𝑗) − (𝑓‘𝑗)) ≤ ((𝑓‘𝑗) − (𝐴‘𝑗)), ((𝐵‘𝑗) − (𝑓‘𝑗)), ((𝑓‘𝑗) − (𝐴‘𝑗)))) = (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝑓‘𝑖)) ≤ ((𝑓‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝑓‘𝑖)), ((𝑓‘𝑖) − (𝐴‘𝑖)))) | 
| 46 | 45 | rneqi 5947 | . . . . . . . 8
⊢ ran
(𝑗 ∈ 𝑋 ↦ if(((𝐵‘𝑗) − (𝑓‘𝑗)) ≤ ((𝑓‘𝑗) − (𝐴‘𝑗)), ((𝐵‘𝑗) − (𝑓‘𝑗)), ((𝑓‘𝑗) − (𝐴‘𝑗)))) = ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝑓‘𝑖)) ≤ ((𝑓‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝑓‘𝑖)), ((𝑓‘𝑖) − (𝐴‘𝑖)))) | 
| 47 | 46 | infeq1i 9519 | . . . . . . 7
⊢ inf(ran
(𝑗 ∈ 𝑋 ↦ if(((𝐵‘𝑗) − (𝑓‘𝑗)) ≤ ((𝑓‘𝑗) − (𝐴‘𝑗)), ((𝐵‘𝑗) − (𝑓‘𝑗)), ((𝑓‘𝑗) − (𝐴‘𝑗)))), ℝ, < ) = inf(ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝑓‘𝑖)) ≤ ((𝑓‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝑓‘𝑖)), ((𝑓‘𝑖) − (𝐴‘𝑖)))), ℝ, < ) | 
| 48 |  | eqid 2736 | . . . . . . 7
⊢ (𝑓(ball‘(𝑎 ∈ (ℝ ↑m 𝑋), 𝑏 ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑘
∈ 𝑋 (((𝑎‘𝑘) − (𝑏‘𝑘))↑2))))inf(ran (𝑗 ∈ 𝑋 ↦ if(((𝐵‘𝑗) − (𝑓‘𝑗)) ≤ ((𝑓‘𝑗) − (𝐴‘𝑗)), ((𝐵‘𝑗) − (𝑓‘𝑗)), ((𝑓‘𝑗) − (𝐴‘𝑗)))), ℝ, < )) = (𝑓(ball‘(𝑎 ∈ (ℝ ↑m 𝑋), 𝑏 ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑘
∈ 𝑋 (((𝑎‘𝑘) − (𝑏‘𝑘))↑2))))inf(ran (𝑗 ∈ 𝑋 ↦ if(((𝐵‘𝑗) − (𝑓‘𝑗)) ≤ ((𝑓‘𝑗) − (𝐴‘𝑗)), ((𝐵‘𝑗) − (𝑓‘𝑗)), ((𝑓‘𝑗) − (𝐴‘𝑗)))), ℝ, < )) | 
| 49 |  | fveq1 6904 | . . . . . . . . . . . 12
⊢ (𝑎 = 𝑔 → (𝑎‘𝑘) = (𝑔‘𝑘)) | 
| 50 | 49 | oveq1d 7447 | . . . . . . . . . . 11
⊢ (𝑎 = 𝑔 → ((𝑎‘𝑘) − (𝑏‘𝑘)) = ((𝑔‘𝑘) − (𝑏‘𝑘))) | 
| 51 | 50 | oveq1d 7447 | . . . . . . . . . 10
⊢ (𝑎 = 𝑔 → (((𝑎‘𝑘) − (𝑏‘𝑘))↑2) = (((𝑔‘𝑘) − (𝑏‘𝑘))↑2)) | 
| 52 | 51 | sumeq2sdv 15740 | . . . . . . . . 9
⊢ (𝑎 = 𝑔 → Σ𝑘 ∈ 𝑋 (((𝑎‘𝑘) − (𝑏‘𝑘))↑2) = Σ𝑘 ∈ 𝑋 (((𝑔‘𝑘) − (𝑏‘𝑘))↑2)) | 
| 53 | 52 | fveq2d 6909 | . . . . . . . 8
⊢ (𝑎 = 𝑔 → (√‘Σ𝑘 ∈ 𝑋 (((𝑎‘𝑘) − (𝑏‘𝑘))↑2)) = (√‘Σ𝑘 ∈ 𝑋 (((𝑔‘𝑘) − (𝑏‘𝑘))↑2))) | 
| 54 |  | fveq1 6904 | . . . . . . . . . . . 12
⊢ (𝑏 = ℎ → (𝑏‘𝑘) = (ℎ‘𝑘)) | 
| 55 | 54 | oveq2d 7448 | . . . . . . . . . . 11
⊢ (𝑏 = ℎ → ((𝑔‘𝑘) − (𝑏‘𝑘)) = ((𝑔‘𝑘) − (ℎ‘𝑘))) | 
| 56 | 55 | oveq1d 7447 | . . . . . . . . . 10
⊢ (𝑏 = ℎ → (((𝑔‘𝑘) − (𝑏‘𝑘))↑2) = (((𝑔‘𝑘) − (ℎ‘𝑘))↑2)) | 
| 57 | 56 | sumeq2sdv 15740 | . . . . . . . . 9
⊢ (𝑏 = ℎ → Σ𝑘 ∈ 𝑋 (((𝑔‘𝑘) − (𝑏‘𝑘))↑2) = Σ𝑘 ∈ 𝑋 (((𝑔‘𝑘) − (ℎ‘𝑘))↑2)) | 
| 58 | 57 | fveq2d 6909 | . . . . . . . 8
⊢ (𝑏 = ℎ → (√‘Σ𝑘 ∈ 𝑋 (((𝑔‘𝑘) − (𝑏‘𝑘))↑2)) = (√‘Σ𝑘 ∈ 𝑋 (((𝑔‘𝑘) − (ℎ‘𝑘))↑2))) | 
| 59 | 53, 58 | cbvmpov 7529 | . . . . . . 7
⊢ (𝑎 ∈ (ℝ
↑m 𝑋),
𝑏 ∈ (ℝ
↑m 𝑋)
↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑎‘𝑘) − (𝑏‘𝑘))↑2))) = (𝑔 ∈ (ℝ ↑m 𝑋), ℎ ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑘
∈ 𝑋 (((𝑔‘𝑘) − (ℎ‘𝑘))↑2))) | 
| 60 | 26, 28, 31, 34, 36, 37, 47, 48, 59 | ioorrnopnlem 46324 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) | 
| 61 | 23, 60 | syldan 591 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) | 
| 62 | 61 | ralrimiva 3145 | . . . 4
⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∀𝑓 ∈ X
𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) | 
| 63 |  | eqid 2736 | . . . . . . . 8
⊢
(TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋)) | 
| 64 | 63 | rrxtop 46309 | . . . . . . 7
⊢ (𝑋 ∈ Fin →
(TopOpen‘(ℝ^‘𝑋)) ∈ Top) | 
| 65 | 24, 64 | syl 17 | . . . . . 6
⊢ (𝜑 →
(TopOpen‘(ℝ^‘𝑋)) ∈ Top) | 
| 66 | 65 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑋 ≠ ∅) →
(TopOpen‘(ℝ^‘𝑋)) ∈ Top) | 
| 67 |  | eltop2 22983 | . . . . 5
⊢
((TopOpen‘(ℝ^‘𝑋)) ∈ Top → (X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈
(TopOpen‘(ℝ^‘𝑋)) ↔ ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))) | 
| 68 | 66, 67 | syl 17 | . . . 4
⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈
(TopOpen‘(ℝ^‘𝑋)) ↔ ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))) | 
| 69 | 62, 68 | mpbird 257 | . . 3
⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈
(TopOpen‘(ℝ^‘𝑋))) | 
| 70 | 16, 69 | syldan 591 | . 2
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈
(TopOpen‘(ℝ^‘𝑋))) | 
| 71 | 14, 70 | pm2.61dan 812 | 1
⊢ (𝜑 → X𝑖 ∈
𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈
(TopOpen‘(ℝ^‘𝑋))) |