Theorem pmltpc 23219

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 2998	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
2	1	rexbii 3041	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
3		rexnal 2995	. . . . . . 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 264	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 ∧ ¬ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
5		rexanali 2998	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ¬ ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))
6	5	rexbii 3041	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 ∧ ¬ (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ∃𝑧 ∈ 𝐴 ¬ ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))
7		rexnal 2995	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 ¬ ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ ¬ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))
8		breq1 4656	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 ≤ 𝑤 ↔ 𝑥 ≤ 𝑤))
9		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
10	9	breq2d 4665	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑤) ≤ (𝐹‘𝑧) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
11	8, 10	imbi12d 334	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑧 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)) ↔ (𝑥 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑥))))
12		breq2 4657	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑦))
13		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
14	13	breq1d 4663	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ((𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
15	12, 14	imbi12d 334	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑥 ≤ 𝑤 → (𝐹‘𝑤) ≤ (𝐹‘𝑥)) ↔ (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))))
16	11, 15	cbvral2v 3179	. . . . . . . 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 324	. . . . . . 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 264	. . . . . 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 733	. . . . 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 3107	. . . . 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 511	. . . . 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 292	. . . 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 3107	. . . . . 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 798	. . . . . . . . . 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 475	. . . . . . . . 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 479	. . . . . . . . 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 799	. . . . . . . . . 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 475	. . . . . . . . 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 800	. . . . . . . . 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 479	. . . . . . . . 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 801	. . . . . . . . 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 802	. . . . . . . . 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 804	. . . . . . . . 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 803	. . . . . . . . 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 805	. . . . . . . . 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 23218	. . . . . . . 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 450	. . . . . . 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 3038	. . . . . 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	syl5bir 233	. . . . 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 3038	. . . 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	syl5bir 233	. . 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 393	. 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 1038	. 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 224	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   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   class class class wbr 4653  dom cdm 5114  ‘cfv 5888  (class class class)co 6650   ↑_pm cpm 7858  ℝcr 9935   < clt 10074   ≤ cle 10075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080

This theorem is referenced by: (None)