Step |
Hyp |
Ref |
Expression |
1 |
|
climadd.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
climadd.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
climadd.4 |
⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 ) |
4 |
|
climle.5 |
⊢ ( 𝜑 → 𝐺 ⇝ 𝐵 ) |
5 |
|
climle.6 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
6 |
|
climle.7 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) ∈ ℝ ) |
7 |
|
climle.8 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐺 ‘ 𝑘 ) ) |
8 |
1
|
fvexi |
⊢ 𝑍 ∈ V |
9 |
8
|
mptex |
⊢ ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) ∈ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) ∈ V ) |
11 |
6
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
12 |
5
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
13 |
|
fveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝐺 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑘 ) ) |
14 |
|
fveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑘 ) ) |
15 |
13 14
|
oveq12d |
⊢ ( 𝑗 = 𝑘 → ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) = ( ( 𝐺 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑘 ) ) ) |
16 |
|
eqid |
⊢ ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) |
17 |
|
ovex |
⊢ ( ( 𝐺 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑘 ) ) ∈ V |
18 |
15 16 17
|
fvmpt |
⊢ ( 𝑘 ∈ 𝑍 → ( ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) ‘ 𝑘 ) = ( ( 𝐺 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑘 ) ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) ‘ 𝑘 ) = ( ( 𝐺 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑘 ) ) ) |
20 |
1 2 4 10 3 11 12 19
|
climsub |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) ⇝ ( 𝐵 − 𝐴 ) ) |
21 |
6 5
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐺 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
22 |
19 21
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) ‘ 𝑘 ) ∈ ℝ ) |
23 |
6 5
|
subge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 0 ≤ ( ( 𝐺 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑘 ) ) ↔ ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐺 ‘ 𝑘 ) ) ) |
24 |
7 23
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 0 ≤ ( ( 𝐺 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑘 ) ) ) |
25 |
24 19
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 0 ≤ ( ( 𝑗 ∈ 𝑍 ↦ ( ( 𝐺 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑗 ) ) ) ‘ 𝑘 ) ) |
26 |
1 2 20 22 25
|
climge0 |
⊢ ( 𝜑 → 0 ≤ ( 𝐵 − 𝐴 ) ) |
27 |
1 2 4 6
|
climrecl |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
28 |
1 2 3 5
|
climrecl |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
29 |
27 28
|
subge0d |
⊢ ( 𝜑 → ( 0 ≤ ( 𝐵 − 𝐴 ) ↔ 𝐴 ≤ 𝐵 ) ) |
30 |
26 29
|
mpbid |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |