Proof of Theorem 2polcon4bN
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl1 1191 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌))) → 𝐾 ∈ HL) | 
| 2 |  | simp1 1136 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝐾 ∈ HL) | 
| 3 |  | 2polss.a | . . . . . . . . 9
⊢ 𝐴 = (Atoms‘𝐾) | 
| 4 |  | 2polss.p | . . . . . . . . 9
⊢  ⊥ =
(⊥𝑃‘𝐾) | 
| 5 | 3, 4 | polssatN 39911 | . . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘𝑌) ⊆ 𝐴) | 
| 6 | 5 | 3adant2 1131 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘𝑌) ⊆ 𝐴) | 
| 7 | 3, 4 | polssatN 39911 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ ( ⊥
‘𝑌) ⊆ 𝐴) → ( ⊥ ‘( ⊥
‘𝑌)) ⊆ 𝐴) | 
| 8 | 2, 6, 7 | syl2anc 584 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥
‘𝑌)) ⊆ 𝐴) | 
| 9 | 8 | adantr 480 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌))) → ( ⊥ ‘( ⊥
‘𝑌)) ⊆ 𝐴) | 
| 10 |  | simpr 484 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌))) → ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌))) | 
| 11 | 3, 4 | polcon3N 39920 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ ( ⊥
‘( ⊥ ‘𝑌)) ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌))) → ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑌))) ⊆ ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑋)))) | 
| 12 | 1, 9, 10, 11 | syl3anc 1372 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌))) → ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑌))) ⊆ ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑋)))) | 
| 13 | 12 | ex 412 | . . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌)) → ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑌))) ⊆ ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑋))))) | 
| 14 | 3, 4 | 3polN 39919 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑌))) = ( ⊥ ‘𝑌)) | 
| 15 | 14 | 3adant2 1131 | . . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑌))) = ( ⊥ ‘𝑌)) | 
| 16 | 3, 4 | 3polN 39919 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) | 
| 17 | 16 | 3adant3 1132 | . . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) | 
| 18 | 15, 17 | sseq12d 4016 | . . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘( ⊥
‘( ⊥ ‘𝑌))) ⊆ ( ⊥ ‘( ⊥
‘( ⊥ ‘𝑋))) ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))) | 
| 19 | 13, 18 | sylibd 239 | . 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌)) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))) | 
| 20 |  | simpl1 1191 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → 𝐾 ∈ HL) | 
| 21 | 3, 4 | polssatN 39911 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) | 
| 22 | 21 | 3adant3 1132 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) | 
| 23 | 22 | adantr 480 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘𝑋) ⊆ 𝐴) | 
| 24 |  | simpr 484 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) | 
| 25 | 3, 4 | polcon3N 39920 | . . . 4
⊢ ((𝐾 ∈ HL ∧ ( ⊥
‘𝑋) ⊆ 𝐴 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌))) | 
| 26 | 20, 23, 24, 25 | syl3anc 1372 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌))) | 
| 27 | 26 | ex 412 | . 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋) → ( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌)))) | 
| 28 | 19, 27 | impbid 212 | 1
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘( ⊥
‘𝑋)) ⊆ ( ⊥
‘( ⊥ ‘𝑌)) ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))) |