Metamath Proof Explorer

Theorem dvgt0

Description: A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015)

Ref Expression
Hypotheses dvgt0.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
dvgt0.b ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
dvgt0.f ⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) )
dvgt0.d ⊢ ( 𝜑 → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℝ+ )
Assertion dvgt0 ( 𝜑 → 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) )

Proof

Step Hyp Ref Expression
1 dvgt0.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 dvgt0.b ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 dvgt0.f ⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) )
4 dvgt0.d ⊢ ( 𝜑 → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℝ+ )
5 ltso ⊢ < Or ℝ
6 1 2 3 4 dvgt0lem1 ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) / ( 𝑦 − 𝑥 ) ) ∈ ℝ+ )
7 6 rpgt0d ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 0 < ( ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) / ( 𝑦 − 𝑥 ) ) )
8 cncff ⊢ ( 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ )
9 3 8 syl ⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ )
10 9 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ )
11 simplrr ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
12 10 11 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ )
13 simplrl ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
14 10 13 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ )
15 12 14 resubcld ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) ∈ ℝ )
16 iccssre ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
17 1 2 16 syl2anc ⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
18 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
19 18 11 sseldd ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ℝ )
20 18 13 sseldd ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ∈ ℝ )
21 19 20 resubcld ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( 𝑦 − 𝑥 ) ∈ ℝ )
22 simpr ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 < 𝑦 )
23 20 19 posdifd ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( 𝑥 < 𝑦 ↔ 0 < ( 𝑦 − 𝑥 ) ) )
24 22 23 mpbid ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 0 < ( 𝑦 − 𝑥 ) )
25 gt0div ⊢ ( ( ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) ∈ ℝ ∧ ( 𝑦 − 𝑥 ) ∈ ℝ ∧ 0 < ( 𝑦 − 𝑥 ) ) → ( 0 < ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) ↔ 0 < ( ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) / ( 𝑦 − 𝑥 ) ) ) )
26 15 21 24 25 syl3anc ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( 0 < ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) ↔ 0 < ( ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) / ( 𝑦 − 𝑥 ) ) ) )
27 7 26 mpbird ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → 0 < ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) )
28 14 12 posdifd ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐹 ‘ 𝑥 ) < ( 𝐹 ‘ 𝑦 ) ↔ 0 < ( ( 𝐹 ‘ 𝑦 ) − ( 𝐹 ‘ 𝑥 ) ) ) )
29 27 28 mpbird ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐹 ‘ 𝑥 ) < ( 𝐹 ‘ 𝑦 ) )
30 1 2 3 4 5 29 dvgt0lem2 ⊢ ( 𝜑 → 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) )