Theorem monotoddzz 37508

Description: A function (given implicitly) 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
monotoddzz.1	⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → 𝐸 < 𝐹))
monotoddzz.2	⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)
monotoddzz.3	⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐺 = -𝐹)
monotoddzz.4	⊢ (𝑥 = 𝐴 → 𝐸 = 𝐶)
monotoddzz.5	⊢ (𝑥 = 𝐵 → 𝐸 = 𝐷)
monotoddzz.6	⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)
monotoddzz.7	⊢ (𝑥 = -𝑦 → 𝐸 = 𝐺)

Assertion

Ref	Expression
monotoddzz	⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ 𝐶 < 𝐷))

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐸   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹   𝑥,𝐺

Allowed substitution hints:   𝐸(𝑥)   𝐹(𝑦)   𝐺(𝑦)

Proof of Theorem monotoddzz

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

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℤ)
2		nffvmpt1 6199	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)
3	2	nfel1 2779	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ
4	1, 3	nfim 1825	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
5		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
6	5	anbi2d 740	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝑎 ∈ ℤ)))
7		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
8	7	eleq1d 2686	. . . . 5 ⊢ (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ))
9	6, 8	imbi12d 334	. . . 4 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)))
10		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
11		monotoddzz.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)
12		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ ℤ ↦ 𝐸) = (𝑥 ∈ ℤ ↦ 𝐸)
13	12	fvmpt2 6291	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝐸 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
14	10, 11, 13	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
15	14, 11	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ)
16	4, 9, 15	chvar 2262	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
17		eleq1 2689	. . . . . 6 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ ℤ ↔ 𝑎 ∈ ℤ))
18	17	anbi2d 740	. . . . 5 ⊢ (𝑦 = 𝑎 → ((𝜑 ∧ 𝑦 ∈ ℤ) ↔ (𝜑 ∧ 𝑎 ∈ ℤ)))
19		negeq 10273	. . . . . . 7 ⊢ (𝑦 = 𝑎 → -𝑦 = -𝑎)
20	19	fveq2d 6195	. . . . . 6 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎))
21		fveq2 6191	. . . . . . 7 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
22	21	negeqd 10275	. . . . . 6 ⊢ (𝑦 = 𝑎 → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
23	20, 22	eqeq12d 2637	. . . . 5 ⊢ (𝑦 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)))
24	18, 23	imbi12d 334	. . . 4 ⊢ (𝑦 = 𝑎 → (((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)) ↔ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))))
25		monotoddzz.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐺 = -𝐹)
26		znegcl 11412	. . . . . . 7 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
27	26	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
28		negex 10279	. . . . . . . 8 ⊢ -𝑦 ∈ V
29		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → (𝑥 ∈ ℤ ↔ -𝑦 ∈ ℤ))
30	29	anbi2d 740	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑦 ∈ ℤ)))
31		monotoddzz.7	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → 𝐸 = 𝐺)
32	31	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (𝐸 ∈ ℝ ↔ 𝐺 ∈ ℝ))
33	30, 32	imbi12d 334	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)))
34	28, 33, 11	vtocl 3259	. . . . . . 7 ⊢ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
35	26, 34	sylan2 491	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
36	31, 12	fvmptg 6280	. . . . . 6 ⊢ ((-𝑦 ∈ ℤ ∧ 𝐺 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = 𝐺)
37	27, 35, 36	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = 𝐺)
38		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
39		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℤ ↔ 𝑦 ∈ ℤ))
40	39	anbi2d 740	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝑦 ∈ ℤ)))
41		monotoddzz.6	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)
42	41	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐸 ∈ ℝ ↔ 𝐹 ∈ ℝ))
43	40, 42	imbi12d 334	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)))
44	43, 11	chvarv 2263	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)
45	41, 12	fvmptg 6280	. . . . . . 7 ⊢ ((𝑦 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
46	38, 44, 45	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
47	46	negeqd 10275	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -𝐹)
48	25, 37, 47	3eqtr4d 2666	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
49	24, 48	chvarv 2263	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
50		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)
51		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 < 𝑏
52		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥 <
53		nffvmpt1 6199	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
54	2, 52, 53	nfbr 4699	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
55	51, 54	nfim 1825	. . . . 5 ⊢ Ⅎ𝑥(𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
56	50, 55	nfim 1825	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
57		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ℕ0 ↔ 𝑎 ∈ ℕ0))
58	57	3anbi2d 1404	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ↔ (𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)))
59		breq1 4656	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 < 𝑏 ↔ 𝑎 < 𝑏))
60	7	breq1d 4663	. . . . . 6 ⊢ (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
61	59, 60	imbi12d 334	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)) ↔ (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))))
62	58, 61	imbi12d 334	. . . 4 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))) ↔ ((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))))
63		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ ℕ0 ↔ 𝑏 ∈ ℕ0))
64	63	3anbi3d 1405	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ↔ (𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)))
65		breq2 4657	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑥 < 𝑦 ↔ 𝑥 < 𝑏))
66		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
67	66	breq2d 4665	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
68	65, 67	imbi12d 334	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)) ↔ (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))))
69	64, 68	imbi12d 334	. . . . 5 ⊢ (𝑦 = 𝑏 → (((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))) ↔ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))))
70		monotoddzz.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → 𝐸 < 𝐹))
71		nn0z 11400	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ)
72	71, 14	sylan2 491	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
73	72	3adant3 1081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
74		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℕ0)
75		nffvmpt1 6199	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)
76	75	nfeq1 2778	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹
77	74, 76	nfim 1825	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
78		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℕ0 ↔ 𝑦 ∈ ℕ0))
79	78	anbi2d 740	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ ℕ0) ↔ (𝜑 ∧ 𝑦 ∈ ℕ0)))
80		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
81	80, 41	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹))
82	79, 81	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸) ↔ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)))
83	77, 82, 72	chvar 2262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
84	83	3adant2 1080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
85	73, 84	breq12d 4666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ 𝐸 < 𝐹))
86	70, 85	sylibrd 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)))
87	69, 86	chvarv 2263	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
88	56, 62, 87	chvar 2262	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
89	16, 49, 88	monotoddzzfi 37507	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵)))
90		simp2 1062	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
91		eleq1 2689	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ℤ ↔ 𝐴 ∈ ℤ))
92	91	anbi2d 740	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝐴 ∈ ℤ)))
93		monotoddzz.4	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐸 = 𝐶)
94	93	eleq1d 2686	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝐶 ∈ ℝ))
95	92, 94	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)))
96	95, 11	vtoclg 3266	. . . . . 6 ⊢ (𝐴 ∈ ℤ → ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ))
97	96	anabsi7 860	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)
98	97	3adant3 1081	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐶 ∈ ℝ)
99	93, 12	fvmptg 6280	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
100	90, 98, 99	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
101		simp3 1063	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
102		eleq1 2689	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ ℤ ↔ 𝐵 ∈ ℤ))
103	102	anbi2d 740	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝐵 ∈ ℤ)))
104		monotoddzz.5	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → 𝐸 = 𝐷)
105	104	eleq1d 2686	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝐷 ∈ ℝ))
106	103, 105	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)))
107	106, 11	vtoclg 3266	. . . . . 6 ⊢ (𝐵 ∈ ℤ → ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ))
108	107	anabsi7 860	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
109	108	3adant2 1080	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
110	104, 12	fvmptg 6280	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
111	101, 109, 110	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
112	100, 111	breq12d 4666	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) ↔ 𝐶 < 𝐷))
113	89, 112	bitrd 268	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ 𝐶 < 𝐷))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  ℝcr 9935   < clt 10074  -cneg 10267  ℕ₀cn0 11292  ℤcz 11377

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-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378

This theorem is referenced by:  ltrmy  37519