Theorem monotoddzz 43650

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 1944	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℤ)
2		nffvmpt1 6894	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)
3	2	nfel1 2941	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ
4	1, 3	nfim 1926	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
5		eleq1 2851	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
6	5	anbi2d 641	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝑎 ∈ ℤ)))
7		fveq2 6883	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
8	7	eleq1d 2848	. . . . 5 ⊢ (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ))
9	6, 8	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)))
10		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
11		monotoddzz.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)
12		eqid 2763	. . . . . . 7 ⊢ (𝑥 ∈ ℤ ↦ 𝐸) = (𝑥 ∈ ℤ ↦ 𝐸)
13	12	fvmpt2 7003	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝐸 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
14	10, 11, 13	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
15	14, 11	eqeltrd 2863	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ)
16	4, 9, 15	chvarfv 2276	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
17		eleq1 2851	. . . . . 6 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ ℤ ↔ 𝑎 ∈ ℤ))
18	17	anbi2d 641	. . . . 5 ⊢ (𝑦 = 𝑎 → ((𝜑 ∧ 𝑦 ∈ ℤ) ↔ (𝜑 ∧ 𝑎 ∈ ℤ)))
19		negeq 11450	. . . . . . 7 ⊢ (𝑦 = 𝑎 → -𝑦 = -𝑎)
20	19	fveq2d 6887	. . . . . 6 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎))
21		fveq2 6883	. . . . . . 7 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
22	21	negeqd 11452	. . . . . 6 ⊢ (𝑦 = 𝑎 → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
23	20, 22	eqeq12d 2779	. . . . 5 ⊢ (𝑦 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)))
24	18, 23	imbi12d 347	. . . 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		monotoddzz.7	. . . . . 6 ⊢ (𝑥 = -𝑦 → 𝐸 = 𝐺)
27		znegcl 12630	. . . . . . 7 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
28	27	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
29		negex 11456	. . . . . . . 8 ⊢ -𝑦 ∈ V
30		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → (𝑥 ∈ ℤ ↔ -𝑦 ∈ ℤ))
31	30	anbi2d 641	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑦 ∈ ℤ)))
32	26	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (𝐸 ∈ ℝ ↔ 𝐺 ∈ ℝ))
33	31, 32	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)))
34	29, 33, 11	vtocl 3526	. . . . . . 7 ⊢ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
35	27, 34	sylan2 604	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
36	12, 26, 28, 35	fvmptd3 7015	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = 𝐺)
37		monotoddzz.6	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)
38		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
39		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℤ ↔ 𝑦 ∈ ℤ))
40	39	anbi2d 641	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝑦 ∈ ℤ)))
41	37	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐸 ∈ ℝ ↔ 𝐹 ∈ ℝ))
42	40, 41	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)))
43	42, 11	chvarvv 2019	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)
44	12, 37, 38, 43	fvmptd3 7015	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
45	44	negeqd 11452	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -𝐹)
46	25, 36, 45	3eqtr4d 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
47	24, 46	chvarvv 2019	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
48		nfv 1944	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)
49		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 < 𝑏
50		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥 <
51		nffvmpt1 6894	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
52	2, 50, 51	nfbr 5159	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
53	49, 52	nfim 1926	. . . . 5 ⊢ Ⅎ𝑥(𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
54	48, 53	nfim 1926	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
55		eleq1 2851	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ℕ0 ↔ 𝑎 ∈ ℕ0))
56	55	3anbi2d 1469	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ↔ (𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)))
57		breq1 5113	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 < 𝑏 ↔ 𝑎 < 𝑏))
58	7	breq1d 5120	. . . . . 6 ⊢ (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
59	57, 58	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)) ↔ (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))))
60	56, 59	imbi12d 347	. . . 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)))))
61		eleq1 2851	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ ℕ0 ↔ 𝑏 ∈ ℕ0))
62	61	3anbi3d 1470	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ↔ (𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)))
63		breq2 5114	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑥 < 𝑦 ↔ 𝑥 < 𝑏))
64		fveq2 6883	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
65	64	breq2d 5122	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
66	63, 65	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)) ↔ (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))))
67	62, 66	imbi12d 347	. . . . 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)))))
68		monotoddzz.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → 𝐸 < 𝐹))
69		nn0z 12616	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ)
70	69, 14	sylan2 604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
71	70	3adant3 1150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
72		nfv 1944	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℕ0)
73		nffvmpt1 6894	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)
74	73	nfeq1 2940	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹
75	72, 74	nfim 1926	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
76		eleq1 2851	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℕ0 ↔ 𝑦 ∈ ℕ0))
77	76	anbi2d 641	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ ℕ0) ↔ (𝜑 ∧ 𝑦 ∈ ℕ0)))
78		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
79	78, 37	eqeq12d 2779	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹))
80	77, 79	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸) ↔ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)))
81	75, 80, 70	chvarfv 2276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
82	81	3adant2 1149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
83	71, 82	breq12d 5123	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ 𝐸 < 𝐹))
84	68, 83	sylibrd 262	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)))
85	67, 84	chvarvv 2019	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
86	54, 60, 85	chvarfv 2276	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
87	16, 47, 86	monotoddzzfi 43649	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵)))
88		monotoddzz.4	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐸 = 𝐶)
89		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
90		eleq1 2851	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ℤ ↔ 𝐴 ∈ ℤ))
91	90	anbi2d 641	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝐴 ∈ ℤ)))
92	88	eleq1d 2848	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝐶 ∈ ℝ))
93	91, 92	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)))
94	93, 11	vtoclg 3523	. . . . . 6 ⊢ (𝐴 ∈ ℤ → ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ))
95	94	anabsi7 683	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)
96	95	3adant3 1150	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐶 ∈ ℝ)
97	12, 88, 89, 96	fvmptd3 7015	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
98		monotoddzz.5	. . . 4 ⊢ (𝑥 = 𝐵 → 𝐸 = 𝐷)
99		simp3 1156	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
100		eleq1 2851	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ ℤ ↔ 𝐵 ∈ ℤ))
101	100	anbi2d 641	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝐵 ∈ ℤ)))
102	98	eleq1d 2848	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝐷 ∈ ℝ))
103	101, 102	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)))
104	103, 11	vtoclg 3523	. . . . . 6 ⊢ (𝐵 ∈ ℤ → ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ))
105	104	anabsi7 683	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
106	105	3adant2 1149	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
107	12, 98, 99, 106	fvmptd3 7015	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
108	97, 107	breq12d 5123	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) ↔ 𝐶 < 𝐷))
109	87, 108	bitrd 282	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ 𝐶 < 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6538  ℝcr 11100   < clt 11244  -cneg 11443  ℕ₀cn0 12505  ℤcz 12592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-n0 12506  df-z 12593

This theorem is referenced by:  ltrmy  43659