Theorem monotoddzzfi 42275

Description: A function which is odd and monotonic on ℕ₀ is monotonic on ℤ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)

Hypotheses

Ref	Expression
monotoddzzfi.1	⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘𝑥) ∈ ℝ)
monotoddzzfi.2	⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘-𝑥) = -(𝐹‘𝑥))
monotoddzzfi.3	⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))

Assertion

Ref	Expression
monotoddzzfi	⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐹‘𝐴) < (𝐹‘𝐵)))

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem monotoddzzfi

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6891	. . 3 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
2		fveq2 6891	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
3		fveq2 6891	. . 3 ⊢ (𝑎 = 𝐵 → (𝐹‘𝑎) = (𝐹‘𝐵))
4		zssre 12581	. . 3 ⊢ ℤ ⊆ ℝ
5		eleq1 2816	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
6	5	anbi2d 628	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝑎 ∈ ℤ)))
7		fveq2 6891	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
8	7	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑎) ∈ ℝ))
9	6, 8	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘𝑥) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ ℤ) → (𝐹‘𝑎) ∈ ℝ)))
10		monotoddzzfi.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘𝑥) ∈ ℝ)
11	9, 10	chvarvv 1995	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ) → (𝐹‘𝑎) ∈ ℝ)
12		elznn 12590	. . . . . . 7 ⊢ (𝑎 ∈ ℤ ↔ (𝑎 ∈ ℝ ∧ (𝑎 ∈ ℕ ∨ -𝑎 ∈ ℕ0)))
13	12	simprbi 496	. . . . . 6 ⊢ (𝑎 ∈ ℤ → (𝑎 ∈ ℕ ∨ -𝑎 ∈ ℕ0))
14		elznn 12590	. . . . . . 7 ⊢ (𝑏 ∈ ℤ ↔ (𝑏 ∈ ℝ ∧ (𝑏 ∈ ℕ ∨ -𝑏 ∈ ℕ0)))
15	14	simprbi 496	. . . . . 6 ⊢ (𝑏 ∈ ℤ → (𝑏 ∈ ℕ ∨ -𝑏 ∈ ℕ0))
16	13, 15	anim12i 612	. . . . 5 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → ((𝑎 ∈ ℕ ∨ -𝑎 ∈ ℕ0) ∧ (𝑏 ∈ ℕ ∨ -𝑏 ∈ ℕ0)))
17	16	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → ((𝑎 ∈ ℕ ∨ -𝑎 ∈ ℕ0) ∧ (𝑏 ∈ ℕ ∨ -𝑏 ∈ ℕ0)))
18		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → 𝜑)
19		nnnn0 12495	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ0)
20	19	ad2antrl 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → 𝑎 ∈ ℕ0)
21		nnnn0 12495	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ0)
22	21	ad2antll 728	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → 𝑏 ∈ ℕ0)
23		vex 3473	. . . . . . . 8 ⊢ 𝑎 ∈ V
24		vex 3473	. . . . . . . 8 ⊢ 𝑏 ∈ V
25		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
26	25	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 ∈ ℕ0 ↔ 𝑎 ∈ ℕ0))
27		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏)
28	27	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑦 ∈ ℕ0 ↔ 𝑏 ∈ ℕ0))
29	26, 28	3anbi23d 1436	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ↔ (𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)))
30		breq12 5147	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 < 𝑦 ↔ 𝑎 < 𝑏))
31		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
32	7, 31	breqan12d 5158	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝐹‘𝑥) < (𝐹‘𝑦) ↔ (𝐹‘𝑎) < (𝐹‘𝑏)))
33	30, 32	imbi12d 344	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)) ↔ (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏))))
34	29, 33	imbi12d 344	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦))) ↔ ((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏)))))
35		monotoddzzfi.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
36	23, 24, 34, 35	vtocl2 3550	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏)))
37	18, 20, 22, 36	syl3anc 1369	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏)))
38	37	ex 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏))))
39	11	adantrr 716	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝐹‘𝑎) ∈ ℝ)
40	39	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (𝐹‘𝑎) ∈ ℝ)
41		0red 11233	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → 0 ∈ ℝ)
42		eleq1 2816	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → (𝑥 ∈ ℤ ↔ 𝑏 ∈ ℤ))
43	42	anbi2d 628	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝑏 ∈ ℤ)))
44		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → (𝐹‘𝑥) = (𝐹‘𝑏))
45	44	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑏) ∈ ℝ))
46	43, 45	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → (((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘𝑥) ∈ ℝ) ↔ ((𝜑 ∧ 𝑏 ∈ ℤ) → (𝐹‘𝑏) ∈ ℝ)))
47	46, 10	chvarvv 1995	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ ℤ) → (𝐹‘𝑏) ∈ ℝ)
48	47	adantrl 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝐹‘𝑏) ∈ ℝ)
49	48	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (𝐹‘𝑏) ∈ ℝ)
50		0red 11233	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → 0 ∈ ℝ)
51		znegcl 12613	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℤ → -𝑎 ∈ ℤ)
52	51	ad2antrl 727	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → -𝑎 ∈ ℤ)
53		negex 11474	. . . . . . . . . . . . . . 15 ⊢ -𝑎 ∈ V
54		eleq1 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = -𝑎 → (𝑥 ∈ ℤ ↔ -𝑎 ∈ ℤ))
55	54	anbi2d 628	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = -𝑎 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑎 ∈ ℤ)))
56		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = -𝑎 → (𝐹‘𝑥) = (𝐹‘-𝑎))
57	56	eleq1d 2813	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = -𝑎 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘-𝑎) ∈ ℝ))
58	55, 57	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = -𝑎 → (((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘𝑥) ∈ ℝ) ↔ ((𝜑 ∧ -𝑎 ∈ ℤ) → (𝐹‘-𝑎) ∈ ℝ)))
59	53, 58, 10	vtocl 3541	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ -𝑎 ∈ ℤ) → (𝐹‘-𝑎) ∈ ℝ)
60	52, 59	syldan 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝐹‘-𝑎) ∈ ℝ)
61	60	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → (𝐹‘-𝑎) ∈ ℝ)
62		0z 12585	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
63		c0ex 11224	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ V
64		eleq1 2816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → (𝑥 ∈ ℤ ↔ 0 ∈ ℤ))
65	64	anbi2d 628	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 0 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 0 ∈ ℤ)))
66		fveq2 6891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘0))
67	66	eleq1d 2813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 0 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘0) ∈ ℝ))
68	65, 67	imbi12d 344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 0 → (((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘𝑥) ∈ ℝ) ↔ ((𝜑 ∧ 0 ∈ ℤ) → (𝐹‘0) ∈ ℝ)))
69	63, 68, 10	vtocl 3541	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 0 ∈ ℤ) → (𝐹‘0) ∈ ℝ)
70	62, 69	mpan2 690	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹‘0) ∈ ℝ)
71	70	recnd 11258	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
72		neg0 11522	. . . . . . . . . . . . . . . . . 18 ⊢ -0 = 0
73	72	fveq2i 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘-0) = (𝐹‘0)
74		negeq 11468	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 0 → -𝑥 = -0)
75	74	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → (𝐹‘-𝑥) = (𝐹‘-0))
76	66	negeqd 11470	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → -(𝐹‘𝑥) = -(𝐹‘0))
77	75, 76	eqeq12d 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 0 → ((𝐹‘-𝑥) = -(𝐹‘𝑥) ↔ (𝐹‘-0) = -(𝐹‘0)))
78	65, 77	imbi12d 344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 0 → (((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘-𝑥) = -(𝐹‘𝑥)) ↔ ((𝜑 ∧ 0 ∈ ℤ) → (𝐹‘-0) = -(𝐹‘0))))
79		monotoddzzfi.2	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘-𝑥) = -(𝐹‘𝑥))
80	63, 78, 79	vtocl 3541	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 0 ∈ ℤ) → (𝐹‘-0) = -(𝐹‘0))
81	62, 80	mpan2 690	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹‘-0) = -(𝐹‘0))
82	73, 81	eqtr3id 2781	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹‘0) = -(𝐹‘0))
83	71, 82	eqnegad 11952	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘0) = 0)
84	83	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝐹‘0) = 0)
85	84	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → (𝐹‘0) = 0)
86		nngt0 12259	. . . . . . . . . . . . . . 15 ⊢ (-𝑎 ∈ ℕ → 0 < -𝑎)
87	86	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → 0 < -𝑎)
88		simplll 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → 𝜑)
89		0nn0 12503	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
90	89	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → 0 ∈ ℕ0)
91		simplrl 776	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → -𝑎 ∈ ℕ0)
92		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → 𝑥 = 0)
93	92	eleq1d 2813	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → (𝑥 ∈ ℕ0 ↔ 0 ∈ ℕ0))
94		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → 𝑦 = -𝑎)
95	94	eleq1d 2813	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → (𝑦 ∈ ℕ0 ↔ -𝑎 ∈ ℕ0))
96	93, 95	3anbi23d 1436	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ↔ (𝜑 ∧ 0 ∈ ℕ0 ∧ -𝑎 ∈ ℕ0)))
97		breq12 5147	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → (𝑥 < 𝑦 ↔ 0 < -𝑎))
98	92	fveq2d 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → (𝐹‘𝑥) = (𝐹‘0))
99	94	fveq2d 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → (𝐹‘𝑦) = (𝐹‘-𝑎))
100	98, 99	breq12d 5155	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → ((𝐹‘𝑥) < (𝐹‘𝑦) ↔ (𝐹‘0) < (𝐹‘-𝑎)))
101	97, 100	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → ((𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)) ↔ (0 < -𝑎 → (𝐹‘0) < (𝐹‘-𝑎))))
102	96, 101	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = -𝑎) → (((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦))) ↔ ((𝜑 ∧ 0 ∈ ℕ0 ∧ -𝑎 ∈ ℕ0) → (0 < -𝑎 → (𝐹‘0) < (𝐹‘-𝑎)))))
103	63, 53, 102, 35	vtocl2 3550	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 0 ∈ ℕ0 ∧ -𝑎 ∈ ℕ0) → (0 < -𝑎 → (𝐹‘0) < (𝐹‘-𝑎)))
104	88, 90, 91, 103	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → (0 < -𝑎 → (𝐹‘0) < (𝐹‘-𝑎)))
105	87, 104	mpd 15	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → (𝐹‘0) < (𝐹‘-𝑎))
106	85, 105	eqbrtrrd 5166	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → 0 < (𝐹‘-𝑎))
107	50, 61, 106	ltled 11378	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 ∈ ℕ) → 0 ≤ (𝐹‘-𝑎))
108		0le0 12329	. . . . . . . . . . . . 13 ⊢ 0 ≤ 0
109	84	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 = 0) → (𝐹‘0) = 0)
110	108, 109	breqtrrid 5180	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 = 0) → 0 ≤ (𝐹‘0))
111		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (-𝑎 = 0 → (𝐹‘-𝑎) = (𝐹‘0))
112	111	breq2d 5154	. . . . . . . . . . . . 13 ⊢ (-𝑎 = 0 → (0 ≤ (𝐹‘-𝑎) ↔ 0 ≤ (𝐹‘0)))
113	112	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 = 0) → (0 ≤ (𝐹‘-𝑎) ↔ 0 ≤ (𝐹‘0)))
114	110, 113	mpbird 257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) ∧ -𝑎 = 0) → 0 ≤ (𝐹‘-𝑎))
115		elnn0 12490	. . . . . . . . . . . . 13 ⊢ (-𝑎 ∈ ℕ0 ↔ (-𝑎 ∈ ℕ ∨ -𝑎 = 0))
116	115	biimpi 215	. . . . . . . . . . . 12 ⊢ (-𝑎 ∈ ℕ0 → (-𝑎 ∈ ℕ ∨ -𝑎 = 0))
117	116	ad2antrl 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (-𝑎 ∈ ℕ ∨ -𝑎 = 0))
118	107, 114, 117	mpjaodan 957	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → 0 ≤ (𝐹‘-𝑎))
119		negeq 11468	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → -𝑥 = -𝑎)
120	119	fveq2d 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝐹‘-𝑥) = (𝐹‘-𝑎))
121	7	negeqd 11470	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → -(𝐹‘𝑥) = -(𝐹‘𝑎))
122	120, 121	eqeq12d 2743	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → ((𝐹‘-𝑥) = -(𝐹‘𝑥) ↔ (𝐹‘-𝑎) = -(𝐹‘𝑎)))
123	6, 122	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘-𝑥) = -(𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑎 ∈ ℤ) → (𝐹‘-𝑎) = -(𝐹‘𝑎))))
124	123, 79	chvarvv 1995	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ) → (𝐹‘-𝑎) = -(𝐹‘𝑎))
125	124	adantrr 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝐹‘-𝑎) = -(𝐹‘𝑎))
126	125	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (𝐹‘-𝑎) = -(𝐹‘𝑎))
127	118, 126	breqtrd 5168	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → 0 ≤ -(𝐹‘𝑎))
128	40	le0neg1d 11801	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → ((𝐹‘𝑎) ≤ 0 ↔ 0 ≤ -(𝐹‘𝑎)))
129	127, 128	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (𝐹‘𝑎) ≤ 0)
130	84	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (𝐹‘0) = 0)
131		nngt0 12259	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → 0 < 𝑏)
132	131	ad2antll 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → 0 < 𝑏)
133		simpll 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → 𝜑)
134	89	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → 0 ∈ ℕ0)
135	21	ad2antll 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → 𝑏 ∈ ℕ0)
136		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → 𝑥 = 0)
137	136	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → (𝑥 ∈ ℕ0 ↔ 0 ∈ ℕ0))
138		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏)
139	138	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → (𝑦 ∈ ℕ0 ↔ 𝑏 ∈ ℕ0))
140	137, 139	3anbi23d 1436	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ↔ (𝜑 ∧ 0 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)))
141		breq12 5147	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → (𝑥 < 𝑦 ↔ 0 < 𝑏))
142	66, 31	breqan12d 5158	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → ((𝐹‘𝑥) < (𝐹‘𝑦) ↔ (𝐹‘0) < (𝐹‘𝑏)))
143	141, 142	imbi12d 344	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → ((𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)) ↔ (0 < 𝑏 → (𝐹‘0) < (𝐹‘𝑏))))
144	140, 143	imbi12d 344	. . . . . . . . . . . 12 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑏) → (((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦))) ↔ ((𝜑 ∧ 0 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (0 < 𝑏 → (𝐹‘0) < (𝐹‘𝑏)))))
145	63, 24, 144, 35	vtocl2 3550	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (0 < 𝑏 → (𝐹‘0) < (𝐹‘𝑏)))
146	133, 134, 135, 145	syl3anc 1369	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (0 < 𝑏 → (𝐹‘0) < (𝐹‘𝑏)))
147	132, 146	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (𝐹‘0) < (𝐹‘𝑏))
148	130, 147	eqbrtrrd 5166	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → 0 < (𝐹‘𝑏))
149	40, 41, 49, 129, 148	lelttrd 11388	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (𝐹‘𝑎) < (𝐹‘𝑏))
150	149	a1d 25	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ)) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏)))
151	150	ex 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → ((-𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏))))
152		simp3 1136	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0) ∧ 𝑎 < 𝑏) → 𝑎 < 𝑏)
153		zre 12578	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℝ)
154	153	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℝ)
155	154	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 𝑏 ∈ ℝ)
156		1red 11231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 1 ∈ ℝ)
157		nnre 12235	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℝ)
158	157	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 𝑎 ∈ ℝ)
159		0red 11233	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 0 ∈ ℝ)
160		nn0ge0 12513	. . . . . . . . . . . . 13 ⊢ (-𝑏 ∈ ℕ0 → 0 ≤ -𝑏)
161	160	ad2antll 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 0 ≤ -𝑏)
162	155	le0neg1d 11801	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → (𝑏 ≤ 0 ↔ 0 ≤ -𝑏))
163	161, 162	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 𝑏 ≤ 0)
164		0le1 11753	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
165	164	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 0 ≤ 1)
166	155, 159, 156, 163, 165	letrd 11387	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 𝑏 ≤ 1)
167		nnge1 12256	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ → 1 ≤ 𝑎)
168	167	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 1 ≤ 𝑎)
169	155, 156, 158, 166, 168	letrd 11387	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → 𝑏 ≤ 𝑎)
170	155, 158	lenltd 11376	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → (𝑏 ≤ 𝑎 ↔ ¬ 𝑎 < 𝑏))
171	169, 170	mpbid 231	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0)) → ¬ 𝑎 < 𝑏)
172	171	3adant3 1130	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0) ∧ 𝑎 < 𝑏) → ¬ 𝑎 < 𝑏)
173	152, 172	pm2.21dd 194	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0) ∧ 𝑎 < 𝑏) → (𝐹‘𝑎) < (𝐹‘𝑏))
174	173	3exp 1117	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → ((𝑎 ∈ ℕ ∧ -𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏))))
175		negex 11474	. . . . . . . . . . . 12 ⊢ -𝑏 ∈ V
176		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → 𝑥 = -𝑏)
177	176	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → (𝑥 ∈ ℕ0 ↔ -𝑏 ∈ ℕ0))
178		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → 𝑦 = -𝑎)
179	178	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → (𝑦 ∈ ℕ0 ↔ -𝑎 ∈ ℕ0))
180	177, 179	3anbi23d 1436	. . . . . . . . . . . . 13 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ↔ (𝜑 ∧ -𝑏 ∈ ℕ0 ∧ -𝑎 ∈ ℕ0)))
181		breq12 5147	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → (𝑥 < 𝑦 ↔ -𝑏 < -𝑎))
182		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = -𝑏 → (𝐹‘𝑥) = (𝐹‘-𝑏))
183		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = -𝑎 → (𝐹‘𝑦) = (𝐹‘-𝑎))
184	182, 183	breqan12d 5158	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → ((𝐹‘𝑥) < (𝐹‘𝑦) ↔ (𝐹‘-𝑏) < (𝐹‘-𝑎)))
185	181, 184	imbi12d 344	. . . . . . . . . . . . 13 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → ((𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)) ↔ (-𝑏 < -𝑎 → (𝐹‘-𝑏) < (𝐹‘-𝑎))))
186	180, 185	imbi12d 344	. . . . . . . . . . . 12 ⊢ ((𝑥 = -𝑏 ∧ 𝑦 = -𝑎) → (((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦))) ↔ ((𝜑 ∧ -𝑏 ∈ ℕ0 ∧ -𝑎 ∈ ℕ0) → (-𝑏 < -𝑎 → (𝐹‘-𝑏) < (𝐹‘-𝑎)))))
187	175, 53, 186, 35	vtocl2 3550	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ -𝑏 ∈ ℕ0 ∧ -𝑎 ∈ ℕ0) → (-𝑏 < -𝑎 → (𝐹‘-𝑏) < (𝐹‘-𝑎)))
188	187	3com23 1124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ -𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0) → (-𝑏 < -𝑎 → (𝐹‘-𝑏) < (𝐹‘-𝑎)))
189	188	3expb 1118	. . . . . . . . 9 ⊢ ((𝜑 ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (-𝑏 < -𝑎 → (𝐹‘-𝑏) < (𝐹‘-𝑎)))
190	189	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (-𝑏 < -𝑎 → (𝐹‘-𝑏) < (𝐹‘-𝑎)))
191		negeq 11468	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑏 → -𝑥 = -𝑏)
192	191	fveq2d 6895	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝐹‘-𝑥) = (𝐹‘-𝑏))
193	44	negeqd 11470	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → -(𝐹‘𝑥) = -(𝐹‘𝑏))
194	192, 193	eqeq12d 2743	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → ((𝐹‘-𝑥) = -(𝐹‘𝑥) ↔ (𝐹‘-𝑏) = -(𝐹‘𝑏)))
195	43, 194	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → (((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘-𝑥) = -(𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑏 ∈ ℤ) → (𝐹‘-𝑏) = -(𝐹‘𝑏))))
196	195, 79	chvarvv 1995	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ ℤ) → (𝐹‘-𝑏) = -(𝐹‘𝑏))
197	196	adantrl 715	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝐹‘-𝑏) = -(𝐹‘𝑏))
198	197	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (𝐹‘-𝑏) = -(𝐹‘𝑏))
199	125	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (𝐹‘-𝑎) = -(𝐹‘𝑎))
200	198, 199	breq12d 5155	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → ((𝐹‘-𝑏) < (𝐹‘-𝑎) ↔ -(𝐹‘𝑏) < -(𝐹‘𝑎)))
201	190, 200	sylibd 238	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (-𝑏 < -𝑎 → -(𝐹‘𝑏) < -(𝐹‘𝑎)))
202		zre 12578	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤ → 𝑎 ∈ ℝ)
203	202	ad2antrl 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → 𝑎 ∈ ℝ)
204	203	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → 𝑎 ∈ ℝ)
205	154	ad2antlr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → 𝑏 ∈ ℝ)
206	204, 205	ltnegd 11808	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (𝑎 < 𝑏 ↔ -𝑏 < -𝑎))
207	39	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (𝐹‘𝑎) ∈ ℝ)
208	48	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (𝐹‘𝑏) ∈ ℝ)
209	207, 208	ltnegd 11808	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → ((𝐹‘𝑎) < (𝐹‘𝑏) ↔ -(𝐹‘𝑏) < -(𝐹‘𝑎)))
210	201, 206, 209	3imtr4d 294	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) ∧ (-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0)) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏)))
211	210	ex 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → ((-𝑎 ∈ ℕ0 ∧ -𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏))))
212	38, 151, 174, 211	ccased 1037	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (((𝑎 ∈ ℕ ∨ -𝑎 ∈ ℕ0) ∧ (𝑏 ∈ ℕ ∨ -𝑏 ∈ ℕ0)) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏))))
213	17, 212	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑎 < 𝑏 → (𝐹‘𝑎) < (𝐹‘𝑏)))
214	1, 2, 3, 4, 11, 213	ltord1 11756	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)) → (𝐴 < 𝐵 ↔ (𝐹‘𝐴) < (𝐹‘𝐵)))
215	214	3impb 1113	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐹‘𝐴) < (𝐹‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  ‘cfv 6542  ℝcr 11123  0cc0 11124  1c1 11125   < clt 11264   ≤ cle 11265  -cneg 11461  ℕcn 12228  ℕ₀cn0 12488  ℤcz 12574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-n0 12489  df-z 12575

This theorem is referenced by:  monotoddzz  42276