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Theorem monotoddzz 38027
 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 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦𝐸 < 𝐹))
monotoddzz.2 ((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)
monotoddzz.3 ((𝜑𝑦 ∈ ℤ) → 𝐺 = -𝐹)
monotoddzz.4 (𝑥 = 𝐴𝐸 = 𝐶)
monotoddzz.5 (𝑥 = 𝐵𝐸 = 𝐷)
monotoddzz.6 (𝑥 = 𝑦𝐸 = 𝐹)
monotoddzz.7 (𝑥 = -𝑦𝐸 = 𝐺)
Assertion
Ref Expression
monotoddzz ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵𝐶 < 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.
StepHypRef Expression
1 nfv 1994 . . . . 5 𝑥(𝜑𝑎 ∈ ℤ)
2 nffvmpt1 6340 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)
32nfel1 2927 . . . . 5 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ
41, 3nfim 1976 . . . 4 𝑥((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
5 eleq1 2837 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
65anbi2d 606 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
7 fveq2 6332 . . . . . 6 (𝑥 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
87eleq1d 2834 . . . . 5 (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ))
96, 8imbi12d 333 . . . 4 (𝑥 = 𝑎 → (((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ) ↔ ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)))
10 simpr 471 . . . . . 6 ((𝜑𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
11 monotoddzz.2 . . . . . 6 ((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)
12 eqid 2770 . . . . . . 7 (𝑥 ∈ ℤ ↦ 𝐸) = (𝑥 ∈ ℤ ↦ 𝐸)
1312fvmpt2 6433 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝐸 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1410, 11, 13syl2anc 565 . . . . 5 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1514, 11eqeltrd 2849 . . . 4 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ)
164, 9, 15chvar 2423 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
17 eleq1 2837 . . . . . 6 (𝑦 = 𝑎 → (𝑦 ∈ ℤ ↔ 𝑎 ∈ ℤ))
1817anbi2d 606 . . . . 5 (𝑦 = 𝑎 → ((𝜑𝑦 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
19 negeq 10474 . . . . . . 7 (𝑦 = 𝑎 → -𝑦 = -𝑎)
2019fveq2d 6336 . . . . . 6 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎))
21 fveq2 6332 . . . . . . 7 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2221negeqd 10476 . . . . . 6 (𝑦 = 𝑎 → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2320, 22eqeq12d 2785 . . . . 5 (𝑦 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)))
2418, 23imbi12d 333 . . . 4 (𝑦 = 𝑎 → (((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)) ↔ ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))))
25 monotoddzz.3 . . . . 5 ((𝜑𝑦 ∈ ℤ) → 𝐺 = -𝐹)
26 znegcl 11613 . . . . . . 7 (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
2726adantl 467 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
28 negex 10480 . . . . . . . 8 -𝑦 ∈ V
29 eleq1 2837 . . . . . . . . . 10 (𝑥 = -𝑦 → (𝑥 ∈ ℤ ↔ -𝑦 ∈ ℤ))
3029anbi2d 606 . . . . . . . . 9 (𝑥 = -𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑦 ∈ ℤ)))
31 monotoddzz.7 . . . . . . . . . 10 (𝑥 = -𝑦𝐸 = 𝐺)
3231eleq1d 2834 . . . . . . . . 9 (𝑥 = -𝑦 → (𝐸 ∈ ℝ ↔ 𝐺 ∈ ℝ))
3330, 32imbi12d 333 . . . . . . . 8 (𝑥 = -𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)))
3428, 33, 11vtocl 3408 . . . . . . 7 ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3526, 34sylan2 572 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3631, 12fvmptg 6422 . . . . . 6 ((-𝑦 ∈ ℤ ∧ 𝐺 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = 𝐺)
3727, 35, 36syl2anc 565 . . . . 5 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = 𝐺)
38 simpr 471 . . . . . . 7 ((𝜑𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
39 eleq1 2837 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ℤ ↔ 𝑦 ∈ ℤ))
4039anbi2d 606 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑦 ∈ ℤ)))
41 monotoddzz.6 . . . . . . . . . 10 (𝑥 = 𝑦𝐸 = 𝐹)
4241eleq1d 2834 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐸 ∈ ℝ ↔ 𝐹 ∈ ℝ))
4340, 42imbi12d 333 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)))
4443, 11chvarv 2424 . . . . . . 7 ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)
4541, 12fvmptg 6422 . . . . . . 7 ((𝑦 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
4638, 44, 45syl2anc 565 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
4746negeqd 10476 . . . . 5 ((𝜑𝑦 ∈ ℤ) → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -𝐹)
4825, 37, 473eqtr4d 2814 . . . 4 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
4924, 48chvarv 2424 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
50 nfv 1994 . . . . 5 𝑥(𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)
51 nfv 1994 . . . . . 6 𝑥 𝑎 < 𝑏
52 nfcv 2912 . . . . . . 7 𝑥 <
53 nffvmpt1 6340 . . . . . . 7 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
542, 52, 53nfbr 4831 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
5551, 54nfim 1976 . . . . 5 𝑥(𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
5650, 55nfim 1976 . . . 4 𝑥((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
57 eleq1 2837 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℕ0𝑎 ∈ ℕ0))
58573anbi2d 1551 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) ↔ (𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)))
59 breq1 4787 . . . . . 6 (𝑥 = 𝑎 → (𝑥 < 𝑏𝑎 < 𝑏))
607breq1d 4794 . . . . . 6 (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
6159, 60imbi12d 333 . . . . 5 (𝑥 = 𝑎 → ((𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)) ↔ (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))))
6258, 61imbi12d 333 . . . 4 (𝑥 = 𝑎 → (((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))) ↔ ((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))))
63 eleq1 2837 . . . . . . 7 (𝑦 = 𝑏 → (𝑦 ∈ ℕ0𝑏 ∈ ℕ0))
64633anbi3d 1552 . . . . . 6 (𝑦 = 𝑏 → ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) ↔ (𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0)))
65 breq2 4788 . . . . . . 7 (𝑦 = 𝑏 → (𝑥 < 𝑦𝑥 < 𝑏))
66 fveq2 6332 . . . . . . . 8 (𝑦 = 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
6766breq2d 4796 . . . . . . 7 (𝑦 = 𝑏 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
6865, 67imbi12d 333 . . . . . 6 (𝑦 = 𝑏 → ((𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)) ↔ (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))))
6964, 68imbi12d 333 . . . . 5 (𝑦 = 𝑏 → (((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))) ↔ ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))))
70 monotoddzz.1 . . . . . 6 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦𝐸 < 𝐹))
71 nn0z 11601 . . . . . . . . 9 (𝑥 ∈ ℕ0𝑥 ∈ ℤ)
7271, 14sylan2 572 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
73723adant3 1125 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
74 nfv 1994 . . . . . . . . . 10 𝑥(𝜑𝑦 ∈ ℕ0)
75 nffvmpt1 6340 . . . . . . . . . . 11 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)
7675nfeq1 2926 . . . . . . . . . 10 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹
7774, 76nfim 1976 . . . . . . . . 9 𝑥((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
78 eleq1 2837 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ℕ0𝑦 ∈ ℕ0))
7978anbi2d 606 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℕ0) ↔ (𝜑𝑦 ∈ ℕ0)))
80 fveq2 6332 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
8180, 41eqeq12d 2785 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹))
8279, 81imbi12d 333 . . . . . . . . 9 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸) ↔ ((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)))
8377, 82, 72chvar 2423 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
84833adant2 1124 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
8573, 84breq12d 4797 . . . . . 6 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ 𝐸 < 𝐹))
8670, 85sylibrd 249 . . . . 5 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)))
8769, 86chvarv 2424 . . . 4 ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
8856, 62, 87chvar 2423 . . 3 ((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
8916, 49, 88monotoddzzfi 38026 . 2 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵)))
90 simp2 1130 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
91 eleq1 2837 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ ℤ ↔ 𝐴 ∈ ℤ))
9291anbi2d 606 . . . . . . . 8 (𝑥 = 𝐴 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐴 ∈ ℤ)))
93 monotoddzz.4 . . . . . . . . 9 (𝑥 = 𝐴𝐸 = 𝐶)
9493eleq1d 2834 . . . . . . . 8 (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝐶 ∈ ℝ))
9592, 94imbi12d 333 . . . . . . 7 (𝑥 = 𝐴 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)))
9695, 11vtoclg 3415 . . . . . 6 (𝐴 ∈ ℤ → ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ))
9796anabsi7 642 . . . . 5 ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)
98973adant3 1125 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐶 ∈ ℝ)
9993, 12fvmptg 6422 . . . 4 ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
10090, 98, 99syl2anc 565 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
101 simp3 1131 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
102 eleq1 2837 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥 ∈ ℤ ↔ 𝐵 ∈ ℤ))
103102anbi2d 606 . . . . . . . 8 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐵 ∈ ℤ)))
104 monotoddzz.5 . . . . . . . . 9 (𝑥 = 𝐵𝐸 = 𝐷)
105104eleq1d 2834 . . . . . . . 8 (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝐷 ∈ ℝ))
106103, 105imbi12d 333 . . . . . . 7 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)))
107106, 11vtoclg 3415 . . . . . 6 (𝐵 ∈ ℤ → ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ))
108107anabsi7 642 . . . . 5 ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
1091083adant2 1124 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
110104, 12fvmptg 6422 . . . 4 ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
111101, 109, 110syl2anc 565 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
112100, 111breq12d 4797 . 2 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) ↔ 𝐶 < 𝐷))
11389, 112bitrd 268 1 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵𝐶 < 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144   class class class wbr 4784   ↦ cmpt 4861  ‘cfv 6031  ℝcr 10136   < clt 10275  -cneg 10468  ℕ0cn0 11493  ℤcz 11578 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-n0 11494  df-z 11579 This theorem is referenced by:  ltrmy  38038
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