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Theorem monotoddzz 42637
Description: A function (given implicitly) which is odd and monotonic on 0 is monotonic on . 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 1910 . . . . 5 𝑥(𝜑𝑎 ∈ ℤ)
2 nffvmpt1 6903 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)
32nfel1 2909 . . . . 5 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ
41, 3nfim 1892 . . . 4 𝑥((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
5 eleq1 2814 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
65anbi2d 628 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
7 fveq2 6892 . . . . . 6 (𝑥 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
87eleq1d 2811 . . . . 5 (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ))
96, 8imbi12d 343 . . . 4 (𝑥 = 𝑎 → (((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ) ↔ ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)))
10 simpr 483 . . . . . 6 ((𝜑𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
11 monotoddzz.2 . . . . . 6 ((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)
12 eqid 2726 . . . . . . 7 (𝑥 ∈ ℤ ↦ 𝐸) = (𝑥 ∈ ℤ ↦ 𝐸)
1312fvmpt2 7011 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝐸 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1410, 11, 13syl2anc 582 . . . . 5 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1514, 11eqeltrd 2826 . . . 4 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ)
164, 9, 15chvarfv 2229 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
17 eleq1 2814 . . . . . 6 (𝑦 = 𝑎 → (𝑦 ∈ ℤ ↔ 𝑎 ∈ ℤ))
1817anbi2d 628 . . . . 5 (𝑦 = 𝑎 → ((𝜑𝑦 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
19 negeq 11492 . . . . . . 7 (𝑦 = 𝑎 → -𝑦 = -𝑎)
2019fveq2d 6896 . . . . . 6 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎))
21 fveq2 6892 . . . . . . 7 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2221negeqd 11494 . . . . . 6 (𝑦 = 𝑎 → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2320, 22eqeq12d 2742 . . . . 5 (𝑦 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)))
2418, 23imbi12d 343 . . . 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 monotoddzz.7 . . . . . 6 (𝑥 = -𝑦𝐸 = 𝐺)
27 znegcl 12642 . . . . . . 7 (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
2827adantl 480 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
29 negex 11498 . . . . . . . 8 -𝑦 ∈ V
30 eleq1 2814 . . . . . . . . . 10 (𝑥 = -𝑦 → (𝑥 ∈ ℤ ↔ -𝑦 ∈ ℤ))
3130anbi2d 628 . . . . . . . . 9 (𝑥 = -𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑦 ∈ ℤ)))
3226eleq1d 2811 . . . . . . . . 9 (𝑥 = -𝑦 → (𝐸 ∈ ℝ ↔ 𝐺 ∈ ℝ))
3331, 32imbi12d 343 . . . . . . . 8 (𝑥 = -𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)))
3429, 33, 11vtocl 3539 . . . . . . 7 ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3527, 34sylan2 591 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3612, 26, 28, 35fvmptd3 7023 . . . . 5 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = 𝐺)
37 monotoddzz.6 . . . . . . 7 (𝑥 = 𝑦𝐸 = 𝐹)
38 simpr 483 . . . . . . 7 ((𝜑𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
39 eleq1 2814 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ℤ ↔ 𝑦 ∈ ℤ))
4039anbi2d 628 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑦 ∈ ℤ)))
4137eleq1d 2811 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐸 ∈ ℝ ↔ 𝐹 ∈ ℝ))
4240, 41imbi12d 343 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)))
4342, 11chvarvv 1995 . . . . . . 7 ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)
4412, 37, 38, 43fvmptd3 7023 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
4544negeqd 11494 . . . . 5 ((𝜑𝑦 ∈ ℤ) → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -𝐹)
4625, 36, 453eqtr4d 2776 . . . 4 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
4724, 46chvarvv 1995 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
48 nfv 1910 . . . . 5 𝑥(𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)
49 nfv 1910 . . . . . 6 𝑥 𝑎 < 𝑏
50 nfcv 2892 . . . . . . 7 𝑥 <
51 nffvmpt1 6903 . . . . . . 7 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
522, 50, 51nfbr 5192 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
5349, 52nfim 1892 . . . . 5 𝑥(𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
5448, 53nfim 1892 . . . 4 𝑥((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
55 eleq1 2814 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℕ0𝑎 ∈ ℕ0))
56553anbi2d 1438 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) ↔ (𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)))
57 breq1 5148 . . . . . 6 (𝑥 = 𝑎 → (𝑥 < 𝑏𝑎 < 𝑏))
587breq1d 5155 . . . . . 6 (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
5957, 58imbi12d 343 . . . . 5 (𝑥 = 𝑎 → ((𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)) ↔ (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))))
6056, 59imbi12d 343 . . . 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)))))
61 eleq1 2814 . . . . . . 7 (𝑦 = 𝑏 → (𝑦 ∈ ℕ0𝑏 ∈ ℕ0))
62613anbi3d 1439 . . . . . 6 (𝑦 = 𝑏 → ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) ↔ (𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0)))
63 breq2 5149 . . . . . . 7 (𝑦 = 𝑏 → (𝑥 < 𝑦𝑥 < 𝑏))
64 fveq2 6892 . . . . . . . 8 (𝑦 = 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
6564breq2d 5157 . . . . . . 7 (𝑦 = 𝑏 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
6663, 65imbi12d 343 . . . . . 6 (𝑦 = 𝑏 → ((𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)) ↔ (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))))
6762, 66imbi12d 343 . . . . 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)))))
68 monotoddzz.1 . . . . . 6 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦𝐸 < 𝐹))
69 nn0z 12628 . . . . . . . . 9 (𝑥 ∈ ℕ0𝑥 ∈ ℤ)
7069, 14sylan2 591 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
71703adant3 1129 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
72 nfv 1910 . . . . . . . . . 10 𝑥(𝜑𝑦 ∈ ℕ0)
73 nffvmpt1 6903 . . . . . . . . . . 11 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)
7473nfeq1 2908 . . . . . . . . . 10 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹
7572, 74nfim 1892 . . . . . . . . 9 𝑥((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
76 eleq1 2814 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ℕ0𝑦 ∈ ℕ0))
7776anbi2d 628 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℕ0) ↔ (𝜑𝑦 ∈ ℕ0)))
78 fveq2 6892 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
7978, 37eqeq12d 2742 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹))
8077, 79imbi12d 343 . . . . . . . . 9 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸) ↔ ((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)))
8175, 80, 70chvarfv 2229 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
82813adant2 1128 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
8371, 82breq12d 5158 . . . . . 6 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ 𝐸 < 𝐹))
8468, 83sylibrd 258 . . . . 5 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)))
8567, 84chvarvv 1995 . . . 4 ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
8654, 60, 85chvarfv 2229 . . 3 ((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
8716, 47, 86monotoddzzfi 42636 . 2 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵)))
88 monotoddzz.4 . . . 4 (𝑥 = 𝐴𝐸 = 𝐶)
89 simp2 1134 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
90 eleq1 2814 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ ℤ ↔ 𝐴 ∈ ℤ))
9190anbi2d 628 . . . . . . . 8 (𝑥 = 𝐴 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐴 ∈ ℤ)))
9288eleq1d 2811 . . . . . . . 8 (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝐶 ∈ ℝ))
9391, 92imbi12d 343 . . . . . . 7 (𝑥 = 𝐴 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)))
9493, 11vtoclg 3535 . . . . . 6 (𝐴 ∈ ℤ → ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ))
9594anabsi7 669 . . . . 5 ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)
96953adant3 1129 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐶 ∈ ℝ)
9712, 88, 89, 96fvmptd3 7023 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
98 monotoddzz.5 . . . 4 (𝑥 = 𝐵𝐸 = 𝐷)
99 simp3 1135 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
100 eleq1 2814 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥 ∈ ℤ ↔ 𝐵 ∈ ℤ))
101100anbi2d 628 . . . . . . . 8 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐵 ∈ ℤ)))
10298eleq1d 2811 . . . . . . . 8 (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝐷 ∈ ℝ))
103101, 102imbi12d 343 . . . . . . 7 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)))
104103, 11vtoclg 3535 . . . . . 6 (𝐵 ∈ ℤ → ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ))
105104anabsi7 669 . . . . 5 ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
1061053adant2 1128 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
10712, 98, 99, 106fvmptd3 7023 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
10897, 107breq12d 5158 . 2 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) ↔ 𝐶 < 𝐷))
10987, 108bitrd 278 1 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵𝐶 < 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099   class class class wbr 5145  cmpt 5228  cfv 6545  cr 11147   < clt 11288  -cneg 11485  0cn0 12517  cz 12603
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 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3968  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8966  df-dom 8967  df-sdom 8968  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12258  df-n0 12518  df-z 12604
This theorem is referenced by:  ltrmy  42646
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