| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq2 7439 | . . . 4
⊢ (𝑡 = 𝑇 → (𝑢 −op 𝑡) = (𝑢 −op 𝑇)) | 
| 2 | 1 | eleq1d 2826 | . . 3
⊢ (𝑡 = 𝑇 → ((𝑢 −op 𝑡) ∈ HrmOp ↔ (𝑢 −op 𝑇) ∈ HrmOp)) | 
| 3 | 1 | fveq1d 6908 | . . . . . 6
⊢ (𝑡 = 𝑇 → ((𝑢 −op 𝑡)‘𝑥) = ((𝑢 −op 𝑇)‘𝑥)) | 
| 4 | 3 | oveq1d 7446 | . . . . 5
⊢ (𝑡 = 𝑇 → (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) = (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥)) | 
| 5 | 4 | breq2d 5155 | . . . 4
⊢ (𝑡 = 𝑇 → (0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥))) | 
| 6 | 5 | ralbidv 3178 | . . 3
⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥))) | 
| 7 | 2, 6 | anbi12d 632 | . 2
⊢ (𝑡 = 𝑇 → (((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥)) ↔ ((𝑢 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥)))) | 
| 8 |  | oveq1 7438 | . . . 4
⊢ (𝑢 = 𝑈 → (𝑢 −op 𝑇) = (𝑈 −op 𝑇)) | 
| 9 | 8 | eleq1d 2826 | . . 3
⊢ (𝑢 = 𝑈 → ((𝑢 −op 𝑇) ∈ HrmOp ↔ (𝑈 −op 𝑇) ∈ HrmOp)) | 
| 10 | 8 | fveq1d 6908 | . . . . . 6
⊢ (𝑢 = 𝑈 → ((𝑢 −op 𝑇)‘𝑥) = ((𝑈 −op 𝑇)‘𝑥)) | 
| 11 | 10 | oveq1d 7446 | . . . . 5
⊢ (𝑢 = 𝑈 → (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) = (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)) | 
| 12 | 11 | breq2d 5155 | . . . 4
⊢ (𝑢 = 𝑈 → (0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))) | 
| 13 | 12 | ralbidv 3178 | . . 3
⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))) | 
| 14 | 9, 13 | anbi12d 632 | . 2
⊢ (𝑢 = 𝑈 → (((𝑢 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥)) ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))) | 
| 15 |  | df-leop 31871 | . 2
⊢ 
≤op = {〈𝑡, 𝑢〉 ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))} | 
| 16 | 7, 14, 15 | brabg 5544 | 1
⊢ ((𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐵) → (𝑇 ≤op 𝑈 ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))) |