Theorem monotoddzz 38974

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 1874	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℤ)
2		nffvmpt1 6507	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)
3	2	nfel1 2939	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ
4	1, 3	nfim 1860	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
5		eleq1 2846	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
6	5	anbi2d 620	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝑎 ∈ ℤ)))
7		fveq2 6496	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
8	7	eleq1d 2843	. . . . 5 ⊢ (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ))
9	6, 8	imbi12d 337	. . . 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 2771	. . . . . . 7 ⊢ (𝑥 ∈ ℤ ↦ 𝐸) = (𝑥 ∈ ℤ ↦ 𝐸)
13	12	fvmpt2 6603	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝐸 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
14	10, 11, 13	syl2anc 576	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
15	14, 11	eqeltrd 2859	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ)
16	4, 9, 15	chvar 2327	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
17		eleq1 2846	. . . . . 6 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ ℤ ↔ 𝑎 ∈ ℤ))
18	17	anbi2d 620	. . . . 5 ⊢ (𝑦 = 𝑎 → ((𝜑 ∧ 𝑦 ∈ ℤ) ↔ (𝜑 ∧ 𝑎 ∈ ℤ)))
19		negeq 10676	. . . . . . 7 ⊢ (𝑦 = 𝑎 → -𝑦 = -𝑎)
20	19	fveq2d 6500	. . . . . 6 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎))
21		fveq2 6496	. . . . . . 7 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
22	21	negeqd 10678	. . . . . 6 ⊢ (𝑦 = 𝑎 → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
23	20, 22	eqeq12d 2786	. . . . 5 ⊢ (𝑦 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)))
24	18, 23	imbi12d 337	. . . 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 11828	. . . . . . 7 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
28	27	adantl 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
29		negex 10682	. . . . . . . 8 ⊢ -𝑦 ∈ V
30		eleq1 2846	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → (𝑥 ∈ ℤ ↔ -𝑦 ∈ ℤ))
31	30	anbi2d 620	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑦 ∈ ℤ)))
32	26	eleq1d 2843	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (𝐸 ∈ ℝ ↔ 𝐺 ∈ ℝ))
33	31, 32	imbi12d 337	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)))
34	29, 33, 11	vtocl 3471	. . . . . . 7 ⊢ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
35	27, 34	sylan2 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
36	12, 26, 28, 35	fvmptd3 6615	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = 𝐺)
37		monotoddzz.6	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)
38		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
39		eleq1 2846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℤ ↔ 𝑦 ∈ ℤ))
40	39	anbi2d 620	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝑦 ∈ ℤ)))
41	37	eleq1d 2843	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐸 ∈ ℝ ↔ 𝐹 ∈ ℝ))
42	40, 41	imbi12d 337	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)))
43	42, 11	chvarv 2328	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)
44	12, 37, 38, 43	fvmptd3 6615	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
45	44	negeqd 10678	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -𝐹)
46	25, 36, 45	3eqtr4d 2817	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
47	24, 46	chvarv 2328	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
48		nfv 1874	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)
49		nfv 1874	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 < 𝑏
50		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥 <
51		nffvmpt1 6507	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
52	2, 50, 51	nfbr 4972	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
53	49, 52	nfim 1860	. . . . 5 ⊢ Ⅎ𝑥(𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
54	48, 53	nfim 1860	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
55		eleq1 2846	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ℕ0 ↔ 𝑎 ∈ ℕ0))
56	55	3anbi2d 1421	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ↔ (𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)))
57		breq1 4928	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 < 𝑏 ↔ 𝑎 < 𝑏))
58	7	breq1d 4935	. . . . . 6 ⊢ (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
59	57, 58	imbi12d 337	. . . . 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 337	. . . 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 2846	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ ℕ0 ↔ 𝑏 ∈ ℕ0))
62	61	3anbi3d 1422	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ↔ (𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)))
63		breq2 4929	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑥 < 𝑦 ↔ 𝑥 < 𝑏))
64		fveq2 6496	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
65	64	breq2d 4937	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
66	63, 65	imbi12d 337	. . . . . 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 337	. . . . 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 11816	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ)
70	69, 14	sylan2 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
71	70	3adant3 1113	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
72		nfv 1874	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℕ0)
73		nffvmpt1 6507	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)
74	73	nfeq1 2938	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹
75	72, 74	nfim 1860	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
76		eleq1 2846	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℕ0 ↔ 𝑦 ∈ ℕ0))
77	76	anbi2d 620	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ ℕ0) ↔ (𝜑 ∧ 𝑦 ∈ ℕ0)))
78		fveq2 6496	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
79	78, 37	eqeq12d 2786	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹))
80	77, 79	imbi12d 337	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸) ↔ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)))
81	75, 80, 70	chvar 2327	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
82	81	3adant2 1112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
83	71, 82	breq12d 4938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ 𝐸 < 𝐹))
84	68, 83	sylibrd 251	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)))
85	67, 84	chvarv 2328	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
86	54, 60, 85	chvar 2327	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
87	16, 47, 86	monotoddzzfi 38973	. 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 1118	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
90		eleq1 2846	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ℤ ↔ 𝐴 ∈ ℤ))
91	90	anbi2d 620	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝐴 ∈ ℤ)))
92	88	eleq1d 2843	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝐶 ∈ ℝ))
93	91, 92	imbi12d 337	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)))
94	93, 11	vtoclg 3479	. . . . . 6 ⊢ (𝐴 ∈ ℤ → ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ))
95	94	anabsi7 659	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)
96	95	3adant3 1113	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐶 ∈ ℝ)
97	12, 88, 89, 96	fvmptd3 6615	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
98		monotoddzz.5	. . . 4 ⊢ (𝑥 = 𝐵 → 𝐸 = 𝐷)
99		simp3 1119	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
100		eleq1 2846	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ ℤ ↔ 𝐵 ∈ ℤ))
101	100	anbi2d 620	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ ℤ) ↔ (𝜑 ∧ 𝐵 ∈ ℤ)))
102	98	eleq1d 2843	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝐷 ∈ ℝ))
103	101, 102	imbi12d 337	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)))
104	103, 11	vtoclg 3479	. . . . . 6 ⊢ (𝐵 ∈ ℤ → ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ))
105	104	anabsi7 659	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
106	105	3adant2 1112	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
107	12, 98, 99, 106	fvmptd3 6615	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
108	97, 107	breq12d 4938	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) ↔ 𝐶 < 𝐷))
109	87, 108	bitrd 271	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ 𝐶 < 𝐷))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   class class class wbr 4925   ↦ cmpt 5004  ‘cfv 6185  ℝcr 10332   < clt 10472  -cneg 10669  ℕ₀cn0 11705  ℤcz 11791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-n0 11706  df-z 11792

This theorem is referenced by:  ltrmy  38983