Proof of Theorem hoiqssbllem1
| Step | Hyp | Ref
| Expression |
| 1 | | hoiqssbllem1.y |
. . . 4
⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m 𝑋)) |
| 2 | 1 | elexd 3504 |
. . 3
⊢ (𝜑 → 𝑌 ∈ V) |
| 3 | | elmapfn 8905 |
. . . 4
⊢ (𝑌 ∈ (ℝ
↑m 𝑋)
→ 𝑌 Fn 𝑋) |
| 4 | 1, 3 | syl 17 |
. . 3
⊢ (𝜑 → 𝑌 Fn 𝑋) |
| 5 | | hoiqssbllem1.i |
. . . 4
⊢
Ⅎ𝑖𝜑 |
| 6 | | hoiqssbllem1.c |
. . . . . . . 8
⊢ (𝜑 → 𝐶:𝑋⟶ℝ) |
| 7 | 6 | ffvelcdmda 7104 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ) |
| 8 | 7 | rexrd 11311 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈
ℝ*) |
| 9 | | hoiqssbllem1.d |
. . . . . . . 8
⊢ (𝜑 → 𝐷:𝑋⟶ℝ) |
| 10 | 9 | ffvelcdmda 7104 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ) |
| 11 | 10 | rexrd 11311 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈
ℝ*) |
| 12 | | elmapi 8889 |
. . . . . . . . 9
⊢ (𝑌 ∈ (ℝ
↑m 𝑋)
→ 𝑌:𝑋⟶ℝ) |
| 13 | 1, 12 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑌:𝑋⟶ℝ) |
| 14 | 13 | ffvelcdmda 7104 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ) |
| 15 | 14 | rexrd 11311 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈
ℝ*) |
| 16 | | hoiqssbllem1.e |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
| 17 | | 2rp 13039 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℝ+ |
| 18 | 17 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 2 ∈
ℝ+) |
| 19 | | hoiqssbllem1.n |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑋 ≠ ∅) |
| 20 | | hoiqssbllem1.x |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑋 ∈ Fin) |
| 21 | | hashnncl 14405 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑋 ∈ Fin →
((♯‘𝑋) ∈
ℕ ↔ 𝑋 ≠
∅)) |
| 22 | 20, 21 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) |
| 23 | 19, 22 | mpbird 257 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (♯‘𝑋) ∈
ℕ) |
| 24 | 23 | nnred 12281 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (♯‘𝑋) ∈
ℝ) |
| 25 | 23 | nngt0d 12315 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 <
(♯‘𝑋)) |
| 26 | 24, 25 | elrpd 13074 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (♯‘𝑋) ∈
ℝ+) |
| 27 | 26 | rpsqrtcld 15450 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(√‘(♯‘𝑋)) ∈
ℝ+) |
| 28 | 18, 27 | rpmulcld 13093 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2 ·
(√‘(♯‘𝑋))) ∈
ℝ+) |
| 29 | 16, 28 | rpdivcld 13094 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈
ℝ+) |
| 30 | 29 | rpred 13077 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈ ℝ) |
| 31 | 30 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈ ℝ) |
| 32 | 14, 31 | resubcld 11691 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ) |
| 33 | 32 | rexrd 11311 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈
ℝ*) |
| 34 | | hoiqssbllem1.l |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) |
| 35 | | iooltub 45523 |
. . . . . . . 8
⊢ ((((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ* ∧
(𝑌‘𝑖) ∈ ℝ* ∧ (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝐶‘𝑖) < (𝑌‘𝑖)) |
| 36 | 33, 15, 34, 35 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝑌‘𝑖)) |
| 37 | 7, 14, 36 | ltled 11409 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ≤ (𝑌‘𝑖)) |
| 38 | 14, 31 | readdcld 11290 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ) |
| 39 | 38 | rexrd 11311 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈
ℝ*) |
| 40 | | hoiqssbllem1.r |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))))) |
| 41 | | ioogtlb 45508 |
. . . . . . 7
⊢ (((𝑌‘𝑖) ∈ ℝ* ∧ ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ* ∧
(𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))))) → (𝑌‘𝑖) < (𝐷‘𝑖)) |
| 42 | 15, 39, 40, 41 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < (𝐷‘𝑖)) |
| 43 | 8, 11, 15, 37, 42 | elicod 13437 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
| 44 | 43 | ex 412 |
. . . 4
⊢ (𝜑 → (𝑖 ∈ 𝑋 → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))) |
| 45 | 5, 44 | ralrimi 3257 |
. . 3
⊢ (𝜑 → ∀𝑖 ∈ 𝑋 (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
| 46 | 2, 4, 45 | 3jca 1129 |
. 2
⊢ (𝜑 → (𝑌 ∈ V ∧ 𝑌 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))) |
| 47 | | elixp2 8941 |
. 2
⊢ (𝑌 ∈ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ↔ (𝑌 ∈ V ∧ 𝑌 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))) |
| 48 | 46, 47 | sylibr 234 |
1
⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |