Theorem pmltpc 25358

Description: Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)

Assertion

Ref	Expression
pmltpc	⊢ ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)) ∨ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))

Distinct variable groups:   𝑎,𝑏,𝑐,𝑥,𝑦,𝐴   𝐹,𝑎,𝑏,𝑐,𝑥,𝑦

Proof of Theorem pmltpc

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexanali 3085	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
2	1	rexbii 3077	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
3		rexnal 3083	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
4	2, 3	bitri 275	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
5		rexanali 3085	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ¬ ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))
6	5	rexbii 3077	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ∃𝑧 ∈ 𝐴 ¬ ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))
7		rexnal 3083	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 ¬ ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ¬ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))
8		breq1 5113	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 ≤ 𝑤 ↔ 𝑥 ≤ 𝑤))
9		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
10	9	breq2d 5122	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑤) ≤ (𝐹‘𝑧) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
11	8, 10	imbi12d 344	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ (𝑥 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑥))))
12		breq2 5114	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑦))
13		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
14	13	breq1d 5120	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ((𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
15	12, 14	imbi12d 344	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑥 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑥)) ↔ (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))))
16	11, 15	cbvral2vw 3220	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
17	7, 16	xchbinx 334	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ¬ ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
18	6, 17	bitri 275	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
19	4, 18	anbi12i 628	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) ↔ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))))
20		reeanv 3210	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))))
21		ioran 985	. . . . 5 ⊢ (¬ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) ↔ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))))
22	19, 20, 21	3bitr4i 303	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) ↔ ¬ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))))
23		reeanv 3210	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) ↔ (∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))))
24		simplll 774	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹))
25	24	simpld 494	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → 𝐹 ∈ (ℝ ↑pm ℝ))
26	24	simprd 495	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → 𝐴 ⊆ dom 𝐹)
27		simpllr 775	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
28	27	simpld 494	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → 𝑥 ∈ 𝐴)
29		simplrl 776	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → 𝑦 ∈ 𝐴)
30	27	simprd 495	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → 𝑧 ∈ 𝐴)
31		simplrr 777	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → 𝑤 ∈ 𝐴)
32		simprll 778	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → 𝑥 ≤ 𝑦)
33		simprrl 780	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → 𝑧 ≤ 𝑤)
34		simprlr 779	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦))
35		simprrr 781	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))
36	25, 26, 28, 29, 30, 31, 32, 33, 34, 35	pmltpclem2 25357	. . . . . . . 8 ⊢ (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
37	36	ex 412	. . . . . . 7 ⊢ ((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
38	37	rexlimdvva 3195	. . . . . 6 ⊢ (((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (∃𝑦 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
39	23, 38	biimtrrid 243	. . . . 5 ⊢ (((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
40	39	rexlimdvva 3195	. . . 4 ⊢ ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∧ ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
41	22, 40	biimtrrid 243	. . 3 ⊢ ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (¬ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
42	41	orrd 863	. 2 ⊢ ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) ∨ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
43		df-3or 1087	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)) ∨ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))) ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) ∨ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
44	42, 43	sylibr 234	1 ⊢ ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)) ∨ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917   class class class wbr 5110  dom cdm 5641  ‘cfv 6514  (class class class)co 7390   ↑_pm cpm 8803  ℝcr 11074   < clt 11215   ≤ cle 11216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-pre-lttri 11149  ax-pre-lttrn 11150

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-er 8674  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221

This theorem is referenced by: (None)