Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  monotoddzz Structured version   Visualization version   GIF version

Theorem monotoddzz 41253
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 1917 . . . . 5 𝑥(𝜑𝑎 ∈ ℤ)
2 nffvmpt1 6853 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎)
32nfel1 2923 . . . . 5 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ
41, 3nfim 1899 . . . 4 𝑥((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
5 eleq1 2825 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℤ ↔ 𝑎 ∈ ℤ))
65anbi2d 629 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
7 fveq2 6842 . . . . . 6 (𝑥 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
87eleq1d 2822 . . . . 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 6959 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝐸 ∈ ℝ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1410, 11, 13syl2anc 584 . . . . 5 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
1514, 11eqeltrd 2838 . . . 4 ((𝜑𝑥 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) ∈ ℝ)
164, 9, 15chvarfv 2233 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) ∈ ℝ)
17 eleq1 2825 . . . . . 6 (𝑦 = 𝑎 → (𝑦 ∈ ℤ ↔ 𝑎 ∈ ℤ))
1817anbi2d 629 . . . . 5 (𝑦 = 𝑎 → ((𝜑𝑦 ∈ ℤ) ↔ (𝜑𝑎 ∈ ℤ)))
19 negeq 11393 . . . . . . 7 (𝑦 = 𝑎 → -𝑦 = -𝑎)
2019fveq2d 6846 . . . . . 6 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎))
21 fveq2 6842 . . . . . . 7 (𝑦 = 𝑎 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
2221negeqd 11395 . . . . . 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 12538 . . . . . . 7 (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
2827adantl 482 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
29 negex 11399 . . . . . . . 8 -𝑦 ∈ V
30 eleq1 2825 . . . . . . . . . 10 (𝑥 = -𝑦 → (𝑥 ∈ ℤ ↔ -𝑦 ∈ ℤ))
3130anbi2d 629 . . . . . . . . 9 (𝑥 = -𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑 ∧ -𝑦 ∈ ℤ)))
3226eleq1d 2822 . . . . . . . . 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 3518 . . . . . . 7 ((𝜑 ∧ -𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3527, 34sylan2 593 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → 𝐺 ∈ ℝ)
3612, 26, 28, 35fvmptd3 6971 . . . . 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 2825 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ℤ ↔ 𝑦 ∈ ℤ))
4039anbi2d 629 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝑦 ∈ ℤ)))
4137eleq1d 2822 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐸 ∈ ℝ ↔ 𝐹 ∈ ℝ))
4240, 41imbi12d 344 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)))
4342, 11chvarvv 2002 . . . . . . 7 ((𝜑𝑦 ∈ ℤ) → 𝐹 ∈ ℝ)
4412, 37, 38, 43fvmptd3 6971 . . . . . 6 ((𝜑𝑦 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
4544negeqd 11395 . . . . 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 2002 . . 3 ((𝜑𝑎 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘-𝑎) = -((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎))
48 nfv 1917 . . . . 5 𝑥(𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)
49 nfv 1917 . . . . . 6 𝑥 𝑎 < 𝑏
50 nfcv 2907 . . . . . . 7 𝑥 <
51 nffvmpt1 6853 . . . . . . 7 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
522, 50, 51nfbr 5152 . . . . . 6 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)
5349, 52nfim 1899 . . . . 5 𝑥(𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
5448, 53nfim 1899 . . . 4 𝑥((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
55 eleq1 2825 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ ℕ0𝑎 ∈ ℕ0))
56553anbi2d 1441 . . . . 5 (𝑥 = 𝑎 → ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) ↔ (𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0)))
57 breq1 5108 . . . . . 6 (𝑥 = 𝑎 → (𝑥 < 𝑏𝑎 < 𝑏))
587breq1d 5115 . . . . . 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 2825 . . . . . . 7 (𝑦 = 𝑏 → (𝑦 ∈ ℕ0𝑏 ∈ ℕ0))
62613anbi3d 1442 . . . . . 6 (𝑦 = 𝑏 → ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) ↔ (𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0)))
63 breq2 5109 . . . . . . 7 (𝑦 = 𝑏 → (𝑥 < 𝑦𝑥 < 𝑏))
64 fveq2 6842 . . . . . . . 8 (𝑦 = 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏))
6564breq2d 5117 . . . . . . 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 12524 . . . . . . . . 9 (𝑥 ∈ ℕ0𝑥 ∈ ℤ)
7069, 14sylan2 593 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
71703adant3 1132 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) = 𝐸)
72 nfv 1917 . . . . . . . . . 10 𝑥(𝜑𝑦 ∈ ℕ0)
73 nffvmpt1 6853 . . . . . . . . . . 11 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦)
7473nfeq1 2922 . . . . . . . . . 10 𝑥((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹
7572, 74nfim 1899 . . . . . . . . 9 𝑥((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
76 eleq1 2825 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ℕ0𝑦 ∈ ℕ0))
7776anbi2d 629 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℕ0) ↔ (𝜑𝑦 ∈ ℕ0)))
78 fveq2 6842 . . . . . . . . . . 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 2233 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
82813adant2 1131 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑦) = 𝐹)
8371, 82breq12d 5118 . . . . . 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 2002 . . . 4 ((𝜑𝑥 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑥 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑥) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
8654, 60, 85chvarfv 2233 . . 3 ((𝜑𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 < 𝑏 → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑎) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝑏)))
8716, 47, 86monotoddzzfi 41252 . 2 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) < ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵)))
88 monotoddzz.4 . . . 4 (𝑥 = 𝐴𝐸 = 𝐶)
89 simp2 1137 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
90 eleq1 2825 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 ∈ ℤ ↔ 𝐴 ∈ ℤ))
9190anbi2d 629 . . . . . . . 8 (𝑥 = 𝐴 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐴 ∈ ℤ)))
9288eleq1d 2822 . . . . . . . 8 (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝐶 ∈ ℝ))
9391, 92imbi12d 344 . . . . . . 7 (𝑥 = 𝐴 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)))
9493, 11vtoclg 3525 . . . . . 6 (𝐴 ∈ ℤ → ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ))
9594anabsi7 669 . . . . 5 ((𝜑𝐴 ∈ ℤ) → 𝐶 ∈ ℝ)
96953adant3 1132 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐶 ∈ ℝ)
9712, 88, 89, 96fvmptd3 6971 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐴) = 𝐶)
98 monotoddzz.5 . . . 4 (𝑥 = 𝐵𝐸 = 𝐷)
99 simp3 1138 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
100 eleq1 2825 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥 ∈ ℤ ↔ 𝐵 ∈ ℤ))
101100anbi2d 629 . . . . . . . 8 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℤ) ↔ (𝜑𝐵 ∈ ℤ)))
10298eleq1d 2822 . . . . . . . 8 (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝐷 ∈ ℝ))
103101, 102imbi12d 344 . . . . . . 7 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) ↔ ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)))
104103, 11vtoclg 3525 . . . . . 6 (𝐵 ∈ ℤ → ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ))
105104anabsi7 669 . . . . 5 ((𝜑𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
1061053adant2 1131 . . . 4 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐷 ∈ ℝ)
10712, 98, 99, 106fvmptd3 6971 . . 3 ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ 𝐸)‘𝐵) = 𝐷)
10897, 107breq12d 5118 . 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 1087   = wceq 1541  wcel 2106   class class class wbr 5105  cmpt 5188  cfv 6496  cr 11050   < clt 11189  -cneg 11386  0cn0 12413  cz 12499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500
This theorem is referenced by:  ltrmy  41262
  Copyright terms: Public domain W3C validator