Theorem pmltpc 25371

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 3084	. . . . . . . 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 3082	. . . . . . 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 3084	. . . . . . . 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 3082	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 ¬ ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ¬ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))
8		breq1 5092	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 ≤ 𝑤 ↔ 𝑥 ≤ 𝑤))
9		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
10	9	breq2d 5101	. . . . . . . . . 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 5093	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑦))
13		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
14	13	breq1d 5099	. . . . . . . . . 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 3212	. . . . . . . 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 3202	. . . . 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 3202	. . . . . 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 25370	. . . . . . . 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 3187	. . . . . 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 3187	. . . 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 2110  ∀wral 3045  ∃wrex 3054   ⊆ wss 3900   class class class wbr 5089  dom cdm 5614  ‘cfv 6477  (class class class)co 7341   ↑_pm cpm 8746  ℝcr 10997   < clt 11138   ≤ cle 11139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-pre-lttri 11072  ax-pre-lttrn 11073

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  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 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-er 8617  df-pm 8748  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144

This theorem is referenced by: (None)