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Theorem monotoddzz 41036
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 1916 . . . . 5 𝑥(𝜑𝑎 ∈ ℤ)
2 nffvmpt1 6836 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)
32nfel1 2920 . . . . 5 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ
41, 3nfim 1898 . . . 4 𝑥((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
5 eleq1 2824 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
65anbi2d 629 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
7 fveq2 6825 . . . . . 6 (𝑥 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
87eleq1d 2821 . . . . 5 (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ))
96, 8imbi12d 344 . . . 4 (𝑥 = 𝑎 → (((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ) ↔ ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)))
10 simpr 485 . . . . . 6 ((𝜑𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
11 monotoddzz.2 . . . . . 6 ((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)
12 eqid 2736 . . . . . . 7 (𝑥 ∈ ℤ ↦ 𝐸) = (𝑥 ∈ ℤ ↦ 𝐸)
1312fvmpt2 6942 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝐸 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1410, 11, 13syl2anc 584 . . . . 5 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1514, 11eqeltrd 2837 . . . 4 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ)
164, 9, 15chvarfv 2232 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
17 eleq1 2824 . . . . . 6 (𝑦 = 𝑎 → (𝑦 ∈ ℤ ↔ 𝑎 ∈ ℤ))
1817anbi2d 629 . . . . 5 (𝑦 = 𝑎 → ((𝜑𝑦 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
19 negeq 11314 . . . . . . 7 (𝑦 = 𝑎 → -𝑦 = -𝑎)
2019fveq2d 6829 . . . . . 6 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎))
21 fveq2 6825 . . . . . . 7 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2221negeqd 11316 . . . . . 6 (𝑦 = 𝑎 → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2320, 22eqeq12d 2752 . . . . 5 (𝑦 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)))
2418, 23imbi12d 344 . . . 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 12456 . . . . . . 7 (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
2827adantl 482 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
29 negex 11320 . . . . . . . 8 -𝑦 ∈ V
30 eleq1 2824 . . . . . . . . . 10 (𝑥 = -𝑦 → (𝑥 ∈ ℤ ↔ -𝑦 ∈ ℤ))
3130anbi2d 629 . . . . . . . . 9 (𝑥 = -𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑦 ∈ ℤ)))
3226eleq1d 2821 . . . . . . . . 9 (𝑥 = -𝑦 → (𝐸 ∈ ℝ ↔ 𝐺 ∈ ℝ))
3331, 32imbi12d 344 . . . . . . . 8 (𝑥 = -𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)))
3429, 33, 11vtocl 3507 . . . . . . 7 ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3527, 34sylan2 593 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3612, 26, 28, 35fvmptd3 6954 . . . . 5 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = 𝐺)
37 monotoddzz.6 . . . . . . 7 (𝑥 = 𝑦𝐸 = 𝐹)
38 simpr 485 . . . . . . 7 ((𝜑𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
39 eleq1 2824 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ℤ ↔ 𝑦 ∈ ℤ))
4039anbi2d 629 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑦 ∈ ℤ)))
4137eleq1d 2821 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐸 ∈ ℝ ↔ 𝐹 ∈ ℝ))
4240, 41imbi12d 344 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)))
4342, 11chvarvv 2001 . . . . . . 7 ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)
4412, 37, 38, 43fvmptd3 6954 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
4544negeqd 11316 . . . . 5 ((𝜑𝑦 ∈ ℤ) → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -𝐹)
4625, 36, 453eqtr4d 2786 . . . 4 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
4724, 46chvarvv 2001 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
48 nfv 1916 . . . . 5 𝑥(𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)
49 nfv 1916 . . . . . 6 𝑥 𝑎 < 𝑏
50 nfcv 2904 . . . . . . 7 𝑥 <
51 nffvmpt1 6836 . . . . . . 7 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
522, 50, 51nfbr 5139 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
5349, 52nfim 1898 . . . . 5 𝑥(𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
5448, 53nfim 1898 . . . 4 𝑥((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
55 eleq1 2824 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℕ0𝑎 ∈ ℕ0))
56553anbi2d 1440 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) ↔ (𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)))
57 breq1 5095 . . . . . 6 (𝑥 = 𝑎 → (𝑥 < 𝑏𝑎 < 𝑏))
587breq1d 5102 . . . . . 6 (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
5957, 58imbi12d 344 . . . . 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 344 . . . 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 2824 . . . . . . 7 (𝑦 = 𝑏 → (𝑦 ∈ ℕ0𝑏 ∈ ℕ0))
62613anbi3d 1441 . . . . . 6 (𝑦 = 𝑏 → ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) ↔ (𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0)))
63 breq2 5096 . . . . . . 7 (𝑦 = 𝑏 → (𝑥 < 𝑦𝑥 < 𝑏))
64 fveq2 6825 . . . . . . . 8 (𝑦 = 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
6564breq2d 5104 . . . . . . 7 (𝑦 = 𝑏 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
6663, 65imbi12d 344 . . . . . 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 344 . . . . 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 12444 . . . . . . . . 9 (𝑥 ∈ ℕ0𝑥 ∈ ℤ)
7069, 14sylan2 593 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
71703adant3 1131 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
72 nfv 1916 . . . . . . . . . 10 𝑥(𝜑𝑦 ∈ ℕ0)
73 nffvmpt1 6836 . . . . . . . . . . 11 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)
7473nfeq1 2919 . . . . . . . . . 10 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹
7572, 74nfim 1898 . . . . . . . . 9 𝑥((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
76 eleq1 2824 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ℕ0𝑦 ∈ ℕ0))
7776anbi2d 629 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℕ0) ↔ (𝜑𝑦 ∈ ℕ0)))
78 fveq2 6825 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
7978, 37eqeq12d 2752 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹))
8077, 79imbi12d 344 . . . . . . . . 9 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸) ↔ ((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)))
8175, 80, 70chvarfv 2232 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
82813adant2 1130 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
8371, 82breq12d 5105 . . . . . 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 2001 . . . 4 ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
8654, 60, 85chvarfv 2232 . . 3 ((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
8716, 47, 86monotoddzzfi 41035 . 2 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵)))
88 monotoddzz.4 . . . 4 (𝑥 = 𝐴𝐸 = 𝐶)
89 simp2 1136 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
90 eleq1 2824 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ ℤ ↔ 𝐴 ∈ ℤ))
9190anbi2d 629 . . . . . . . 8 (𝑥 = 𝐴 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐴 ∈ ℤ)))
9288eleq1d 2821 . . . . . . . 8 (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝐶 ∈ ℝ))
9391, 92imbi12d 344 . . . . . . 7 (𝑥 = 𝐴 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)))
9493, 11vtoclg 3514 . . . . . 6 (𝐴 ∈ ℤ → ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ))
9594anabsi7 668 . . . . 5 ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)
96953adant3 1131 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐶 ∈ ℝ)
9712, 88, 89, 96fvmptd3 6954 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
98 monotoddzz.5 . . . 4 (𝑥 = 𝐵𝐸 = 𝐷)
99 simp3 1137 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
100 eleq1 2824 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥 ∈ ℤ ↔ 𝐵 ∈ ℤ))
101100anbi2d 629 . . . . . . . 8 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐵 ∈ ℤ)))
10298eleq1d 2821 . . . . . . . 8 (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝐷 ∈ ℝ))
103101, 102imbi12d 344 . . . . . . 7 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)))
104103, 11vtoclg 3514 . . . . . 6 (𝐵 ∈ ℤ → ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ))
105104anabsi7 668 . . . . 5 ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
1061053adant2 1130 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
10712, 98, 99, 106fvmptd3 6954 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
10897, 107breq12d 5105 . 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 396  w3a 1086   = wceq 1540  wcel 2105   class class class wbr 5092  cmpt 5175  cfv 6479  cr 10971   < clt 11110  -cneg 11307  0cn0 12334  cz 12420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-n0 12335  df-z 12421
This theorem is referenced by:  ltrmy  41045
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