Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → 𝐴 ∈ ℝ ) |
2 |
|
hmopd |
⊢ ( ( 𝑈 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ) → ( 𝑈 −op 𝑇 ) ∈ HrmOp ) |
3 |
2
|
ancoms |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( 𝑈 −op 𝑇 ) ∈ HrmOp ) |
4 |
3
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( 𝑈 −op 𝑇 ) ∈ HrmOp ) |
5 |
|
leopmuli |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 𝑈 −op 𝑇 ) ∈ HrmOp ) ∧ ( 0 ≤ 𝐴 ∧ 0hop ≤op ( 𝑈 −op 𝑇 ) ) ) → 0hop ≤op ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) ) |
6 |
5
|
exp32 |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝑈 −op 𝑇 ) ∈ HrmOp ) → ( 0 ≤ 𝐴 → ( 0hop ≤op ( 𝑈 −op 𝑇 ) → 0hop ≤op ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) ) ) ) |
7 |
1 4 6
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( 0 ≤ 𝐴 → ( 0hop ≤op ( 𝑈 −op 𝑇 ) → 0hop ≤op ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) ) ) ) |
8 |
7
|
imp |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) ∧ 0 ≤ 𝐴 ) → ( 0hop ≤op ( 𝑈 −op 𝑇 ) → 0hop ≤op ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) ) ) |
9 |
|
leop3 |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( 𝑇 ≤op 𝑈 ↔ 0hop ≤op ( 𝑈 −op 𝑇 ) ) ) |
10 |
9
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( 𝑇 ≤op 𝑈 ↔ 0hop ≤op ( 𝑈 −op 𝑇 ) ) ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) ∧ 0 ≤ 𝐴 ) → ( 𝑇 ≤op 𝑈 ↔ 0hop ≤op ( 𝑈 −op 𝑇 ) ) ) |
12 |
|
hmopm |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ) → ( 𝐴 ·op 𝑇 ) ∈ HrmOp ) |
13 |
|
hmopm |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑈 ∈ HrmOp ) → ( 𝐴 ·op 𝑈 ) ∈ HrmOp ) |
14 |
|
leop3 |
⊢ ( ( ( 𝐴 ·op 𝑇 ) ∈ HrmOp ∧ ( 𝐴 ·op 𝑈 ) ∈ HrmOp ) → ( ( 𝐴 ·op 𝑇 ) ≤op ( 𝐴 ·op 𝑈 ) ↔ 0hop ≤op ( ( 𝐴 ·op 𝑈 ) −op ( 𝐴 ·op 𝑇 ) ) ) ) |
15 |
12 13 14
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ) ∧ ( 𝐴 ∈ ℝ ∧ 𝑈 ∈ HrmOp ) ) → ( ( 𝐴 ·op 𝑇 ) ≤op ( 𝐴 ·op 𝑈 ) ↔ 0hop ≤op ( ( 𝐴 ·op 𝑈 ) −op ( 𝐴 ·op 𝑇 ) ) ) ) |
16 |
15
|
3impdi |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( ( 𝐴 ·op 𝑇 ) ≤op ( 𝐴 ·op 𝑈 ) ↔ 0hop ≤op ( ( 𝐴 ·op 𝑈 ) −op ( 𝐴 ·op 𝑇 ) ) ) ) |
17 |
|
recn |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ ) |
18 |
|
hmopf |
⊢ ( 𝑈 ∈ HrmOp → 𝑈 : ℋ ⟶ ℋ ) |
19 |
|
hmopf |
⊢ ( 𝑇 ∈ HrmOp → 𝑇 : ℋ ⟶ ℋ ) |
20 |
|
hosubdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑈 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) = ( ( 𝐴 ·op 𝑈 ) −op ( 𝐴 ·op 𝑇 ) ) ) |
21 |
17 18 19 20
|
syl3an |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑈 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ) → ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) = ( ( 𝐴 ·op 𝑈 ) −op ( 𝐴 ·op 𝑇 ) ) ) |
22 |
21
|
3com23 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) = ( ( 𝐴 ·op 𝑈 ) −op ( 𝐴 ·op 𝑇 ) ) ) |
23 |
22
|
breq2d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( 0hop ≤op ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) ↔ 0hop ≤op ( ( 𝐴 ·op 𝑈 ) −op ( 𝐴 ·op 𝑇 ) ) ) ) |
24 |
16 23
|
bitr4d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) → ( ( 𝐴 ·op 𝑇 ) ≤op ( 𝐴 ·op 𝑈 ) ↔ 0hop ≤op ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) ) ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) ∧ 0 ≤ 𝐴 ) → ( ( 𝐴 ·op 𝑇 ) ≤op ( 𝐴 ·op 𝑈 ) ↔ 0hop ≤op ( 𝐴 ·op ( 𝑈 −op 𝑇 ) ) ) ) |
26 |
8 11 25
|
3imtr4d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) ∧ 0 ≤ 𝐴 ) → ( 𝑇 ≤op 𝑈 → ( 𝐴 ·op 𝑇 ) ≤op ( 𝐴 ·op 𝑈 ) ) ) |
27 |
26
|
impr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ) ∧ ( 0 ≤ 𝐴 ∧ 𝑇 ≤op 𝑈 ) ) → ( 𝐴 ·op 𝑇 ) ≤op ( 𝐴 ·op 𝑈 ) ) |