Proof of Theorem eliccelioc
| Step | Hyp | Ref
| Expression |
| 1 | | iocssicc 13477 |
. . . . 5
⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) |
| 2 | 1 | sseli 3979 |
. . . 4
⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐶 ∈ (𝐴[,]𝐵)) |
| 3 | 2 | adantl 481 |
. . 3
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ (𝐴[,]𝐵)) |
| 4 | | eliccelioc.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 5 | 4 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 ∈ ℝ) |
| 6 | 4 | rexrd 11311 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 7 | 6 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 ∈
ℝ*) |
| 8 | | eliccelioc.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 9 | 8 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐵 ∈ ℝ) |
| 10 | 9 | rexrd 11311 |
. . . . 5
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐵 ∈
ℝ*) |
| 11 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ (𝐴(,]𝐵)) |
| 12 | | iocgtlb 45515 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐶
∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶) |
| 13 | 7, 10, 11, 12 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶) |
| 14 | 5, 13 | gtned 11396 |
. . 3
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≠ 𝐴) |
| 15 | 3, 14 | jca 511 |
. 2
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) |
| 16 | 6 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐴 ∈
ℝ*) |
| 17 | 8 | rexrd 11311 |
. . . 4
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 18 | 17 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐵 ∈
ℝ*) |
| 19 | | eliccelioc.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈
ℝ*) |
| 20 | 19 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐶 ∈
ℝ*) |
| 21 | 4 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐴 ∈ ℝ) |
| 22 | 4, 8 | iccssred 13474 |
. . . . . 6
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 23 | 22 | sselda 3983 |
. . . . 5
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ) |
| 24 | 23 | adantrr 717 |
. . . 4
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐶 ∈ ℝ) |
| 25 | 6 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ∈
ℝ*) |
| 26 | 17 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐵 ∈
ℝ*) |
| 27 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ (𝐴[,]𝐵)) |
| 28 | | iccgelb 13443 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐶
∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) |
| 29 | 25, 26, 27, 28 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) |
| 30 | 29 | adantrr 717 |
. . . 4
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐴 ≤ 𝐶) |
| 31 | | simprr 773 |
. . . 4
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐶 ≠ 𝐴) |
| 32 | 21, 24, 30, 31 | leneltd 11415 |
. . 3
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐴 < 𝐶) |
| 33 | | iccleub 13442 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐶
∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) |
| 34 | 25, 26, 27, 33 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) |
| 35 | 34 | adantrr 717 |
. . 3
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐶 ≤ 𝐵) |
| 36 | 16, 18, 20, 32, 35 | eliocd 45520 |
. 2
⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)) → 𝐶 ∈ (𝐴(,]𝐵)) |
| 37 | 15, 36 | impbida 801 |
1
⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴))) |