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Theorem monotoddzz 39418
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 1906 . . . . 5 𝑥(𝜑𝑎 ∈ ℤ)
2 nffvmpt1 6674 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)
32nfel1 2991 . . . . 5 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ
41, 3nfim 1888 . . . 4 𝑥((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
5 eleq1 2897 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
65anbi2d 628 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
7 fveq2 6663 . . . . . 6 (𝑥 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
87eleq1d 2894 . . . . 5 (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ))
96, 8imbi12d 346 . . . 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 2818 . . . . . . 7 (𝑥 ∈ ℤ ↦ 𝐸) = (𝑥 ∈ ℤ ↦ 𝐸)
1312fvmpt2 6771 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝐸 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1410, 11, 13syl2anc 584 . . . . 5 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1514, 11eqeltrd 2910 . . . 4 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ)
164, 9, 15chvarfv 2232 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
17 eleq1 2897 . . . . . 6 (𝑦 = 𝑎 → (𝑦 ∈ ℤ ↔ 𝑎 ∈ ℤ))
1817anbi2d 628 . . . . 5 (𝑦 = 𝑎 → ((𝜑𝑦 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
19 negeq 10866 . . . . . . 7 (𝑦 = 𝑎 → -𝑦 = -𝑎)
2019fveq2d 6667 . . . . . 6 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎))
21 fveq2 6663 . . . . . . 7 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2221negeqd 10868 . . . . . 6 (𝑦 = 𝑎 → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2320, 22eqeq12d 2834 . . . . 5 (𝑦 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)))
2418, 23imbi12d 346 . . . 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 12005 . . . . . . 7 (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
2827adantl 482 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
29 negex 10872 . . . . . . . 8 -𝑦 ∈ V
30 eleq1 2897 . . . . . . . . . 10 (𝑥 = -𝑦 → (𝑥 ∈ ℤ ↔ -𝑦 ∈ ℤ))
3130anbi2d 628 . . . . . . . . 9 (𝑥 = -𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑦 ∈ ℤ)))
3226eleq1d 2894 . . . . . . . . 9 (𝑥 = -𝑦 → (𝐸 ∈ ℝ ↔ 𝐺 ∈ ℝ))
3331, 32imbi12d 346 . . . . . . . 8 (𝑥 = -𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)))
3429, 33, 11vtocl 3557 . . . . . . 7 ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3527, 34sylan2 592 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3612, 26, 28, 35fvmptd3 6783 . . . . 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 2897 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ℤ ↔ 𝑦 ∈ ℤ))
4039anbi2d 628 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑦 ∈ ℤ)))
4137eleq1d 2894 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐸 ∈ ℝ ↔ 𝐹 ∈ ℝ))
4240, 41imbi12d 346 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)))
4342, 11chvarvv 1996 . . . . . . 7 ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)
4412, 37, 38, 43fvmptd3 6783 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
4544negeqd 10868 . . . . 5 ((𝜑𝑦 ∈ ℤ) → -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = -𝐹)
4625, 36, 453eqtr4d 2863 . . . 4 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
4724, 46chvarvv 1996 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
48 nfv 1906 . . . . 5 𝑥(𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)
49 nfv 1906 . . . . . 6 𝑥 𝑎 < 𝑏
50 nfcv 2974 . . . . . . 7 𝑥 <
51 nffvmpt1 6674 . . . . . . 7 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
522, 50, 51nfbr 5104 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
5349, 52nfim 1888 . . . . 5 𝑥(𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
5448, 53nfim 1888 . . . 4 𝑥((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
55 eleq1 2897 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℕ0𝑎 ∈ ℕ0))
56553anbi2d 1432 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) ↔ (𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)))
57 breq1 5060 . . . . . 6 (𝑥 = 𝑎 → (𝑥 < 𝑏𝑎 < 𝑏))
587breq1d 5067 . . . . . 6 (𝑥 = 𝑎 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
5957, 58imbi12d 346 . . . . 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 346 . . . 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 2897 . . . . . . 7 (𝑦 = 𝑏 → (𝑦 ∈ ℕ0𝑏 ∈ ℕ0))
62613anbi3d 1433 . . . . . 6 (𝑦 = 𝑏 → ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) ↔ (𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0)))
63 breq2 5061 . . . . . . 7 (𝑦 = 𝑏 → (𝑥 < 𝑦𝑥 < 𝑏))
64 fveq2 6663 . . . . . . . 8 (𝑦 = 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
6564breq2d 5069 . . . . . . 7 (𝑦 = 𝑏 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
6663, 65imbi12d 346 . . . . . 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 346 . . . . 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 11993 . . . . . . . . 9 (𝑥 ∈ ℕ0𝑥 ∈ ℤ)
7069, 14sylan2 592 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
71703adant3 1124 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
72 nfv 1906 . . . . . . . . . 10 𝑥(𝜑𝑦 ∈ ℕ0)
73 nffvmpt1 6674 . . . . . . . . . . 11 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)
7473nfeq1 2990 . . . . . . . . . 10 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹
7572, 74nfim 1888 . . . . . . . . 9 𝑥((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
76 eleq1 2897 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ℕ0𝑦 ∈ ℕ0))
7776anbi2d 628 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℕ0) ↔ (𝜑𝑦 ∈ ℕ0)))
78 fveq2 6663 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦))
7978, 37eqeq12d 2834 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹))
8077, 79imbi12d 346 . . . . . . . . 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 1123 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
8371, 82breq12d 5070 . . . . . 6 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) ↔ 𝐸 < 𝐹))
8468, 83sylibrd 260 . . . . 5 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)))
8567, 84chvarvv 1996 . . . 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 39417 . 2 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵)))
88 monotoddzz.4 . . . 4 (𝑥 = 𝐴𝐸 = 𝐶)
89 simp2 1129 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
90 eleq1 2897 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ ℤ ↔ 𝐴 ∈ ℤ))
9190anbi2d 628 . . . . . . . 8 (𝑥 = 𝐴 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐴 ∈ ℤ)))
9288eleq1d 2894 . . . . . . . 8 (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝐶 ∈ ℝ))
9391, 92imbi12d 346 . . . . . . 7 (𝑥 = 𝐴 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)))
9493, 11vtoclg 3565 . . . . . 6 (𝐴 ∈ ℤ → ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ))
9594anabsi7 667 . . . . 5 ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)
96953adant3 1124 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐶 ∈ ℝ)
9712, 88, 89, 96fvmptd3 6783 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
98 monotoddzz.5 . . . 4 (𝑥 = 𝐵𝐸 = 𝐷)
99 simp3 1130 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
100 eleq1 2897 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥 ∈ ℤ ↔ 𝐵 ∈ ℤ))
101100anbi2d 628 . . . . . . . 8 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐵 ∈ ℤ)))
10298eleq1d 2894 . . . . . . . 8 (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝐷 ∈ ℝ))
103101, 102imbi12d 346 . . . . . . 7 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)))
104103, 11vtoclg 3565 . . . . . 6 (𝐵 ∈ ℤ → ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ))
105104anabsi7 667 . . . . 5 ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
1061053adant2 1123 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
10712, 98, 99, 106fvmptd3 6783 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
10897, 107breq12d 5070 . 2 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) ↔ 𝐶 < 𝐷))
10987, 108bitrd 280 1 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵𝐶 < 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cmpt 5137  cfv 6348  cr 10524   < clt 10663  -cneg 10859  0cn0 11885  cz 11969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970
This theorem is referenced by:  ltrmy  39427
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