| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | dib11.b | . . . . 5
⊢ 𝐵 = (Base‘𝐾) | 
| 2 |  | dib11.l | . . . . 5
⊢  ≤ =
(le‘𝐾) | 
| 3 |  | dib11.h | . . . . 5
⊢ 𝐻 = (LHyp‘𝐾) | 
| 4 |  | eqid 2736 | . . . . 5
⊢
((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | 
| 5 |  | eqid 2736 | . . . . 5
⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) | 
| 6 |  | eqid 2736 | . . . . 5
⊢
((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | 
| 7 |  | dib11.i | . . . . 5
⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | 
| 8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 41147 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})) | 
| 9 | 8 | 3adant3 1132 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})) | 
| 10 | 1, 2, 3, 4, 5, 6, 7 | dibval2 41147 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘𝑌) = ((((DIsoA‘𝐾)‘𝑊)‘𝑌) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})) | 
| 11 | 10 | 3adant2 1131 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘𝑌) = ((((DIsoA‘𝐾)‘𝑊)‘𝑌) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})) | 
| 12 | 9, 11 | sseq12d 4016 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ⊆ ((((DIsoA‘𝐾)‘𝑊)‘𝑌) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))) | 
| 13 | 1, 2, 3, 7 | dibn0 41156 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) | 
| 14 | 13 | 3adant3 1132 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) | 
| 15 | 9, 14 | eqnetrrd 3008 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ≠ ∅) | 
| 16 |  | ssxpb 6193 | . . 3
⊢
(((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ≠ ∅ →
(((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ⊆ ((((DIsoA‘𝐾)‘𝑊)‘𝑌) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ↔ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ∧ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ⊆ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))) | 
| 17 | 15, 16 | syl 17 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ⊆ ((((DIsoA‘𝐾)‘𝑊)‘𝑌) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ↔ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ∧ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ⊆ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))) | 
| 18 |  | ssid 4005 | . . . 4
⊢ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ⊆ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} | 
| 19 | 18 | biantru 529 | . . 3
⊢
((((DIsoA‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ↔ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ∧ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ⊆ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})) | 
| 20 | 1, 2, 3, 6 | diaord 41050 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((((DIsoA‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ↔ 𝑋 ≤ 𝑌)) | 
| 21 | 19, 20 | bitr3id 285 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (((((DIsoA‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ∧ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ⊆ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ↔ 𝑋 ≤ 𝑌)) | 
| 22 | 12, 17, 21 | 3bitrd 305 | 1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |