Proof of Theorem cmtbr4N
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | cmtbr4.b | . . 3
⊢ 𝐵 = (Base‘𝐾) | 
| 2 |  | cmtbr4.j | . . 3
⊢  ∨ =
(join‘𝐾) | 
| 3 |  | cmtbr4.m | . . 3
⊢  ∧ =
(meet‘𝐾) | 
| 4 |  | cmtbr4.o | . . 3
⊢  ⊥ =
(oc‘𝐾) | 
| 5 |  | cmtbr4.c | . . 3
⊢ 𝐶 = (cm‘𝐾) | 
| 6 | 1, 2, 3, 4, 5 | cmtbr3N 39256 | . 2
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌))) | 
| 7 |  | omllat 39244 | . . . . 5
⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | 
| 8 |  | cmtbr4.l | . . . . . 6
⊢  ≤ =
(le‘𝐾) | 
| 9 | 1, 8, 3 | latmle2 18511 | . . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌) | 
| 10 | 7, 9 | syl3an1 1163 | . . . 4
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌) | 
| 11 |  | breq1 5145 | . . . 4
⊢ ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) ≤ 𝑌)) | 
| 12 | 10, 11 | syl5ibrcom 247 | . . 3
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌)) | 
| 13 | 7 | 3ad2ant1 1133 | . . . . . . . . 9
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | 
| 14 |  | simp2 1137 | . . . . . . . . 9
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | 
| 15 |  | omlop 39243 | . . . . . . . . . . . 12
⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | 
| 16 | 15 | 3ad2ant1 1133 | . . . . . . . . . . 11
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP) | 
| 17 | 1, 4 | opoccl 39196 | . . . . . . . . . . 11
⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) | 
| 18 | 16, 14, 17 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) | 
| 19 |  | simp3 1138 | . . . . . . . . . 10
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | 
| 20 | 1, 2 | latjcl 18485 | . . . . . . . . . 10
⊢ ((𝐾 ∈ Lat ∧ ( ⊥
‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) | 
| 21 | 13, 18, 19, 20 | syl3anc 1372 | . . . . . . . . 9
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) | 
| 22 | 1, 8, 3 | latmle1 18510 | . . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋) | 
| 23 | 13, 14, 21, 22 | syl3anc 1372 | . . . . . . . 8
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋) | 
| 24 | 23 | anim1i 615 | . . . . . . 7
⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌)) | 
| 25 | 24 | ex 412 | . . . . . 6
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))) | 
| 26 | 1, 3 | latmcl 18486 | . . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵) | 
| 27 | 13, 14, 21, 26 | syl3anc 1372 | . . . . . . 7
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵) | 
| 28 | 1, 8, 3 | latlem12 18512 | . . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌))) | 
| 29 | 13, 27, 14, 19, 28 | syl13anc 1373 | . . . . . 6
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌))) | 
| 30 | 25, 29 | sylibd 239 | . . . . 5
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌))) | 
| 31 | 1, 8, 2 | latlej2 18495 | . . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ ( ⊥
‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌)) | 
| 32 | 13, 18, 19, 31 | syl3anc 1372 | . . . . . 6
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌)) | 
| 33 | 1, 8, 3 | latmlem2 18516 | . . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑌 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))) | 
| 34 | 13, 19, 21, 14, 33 | syl13anc 1373 | . . . . . 6
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))) | 
| 35 | 32, 34 | mpd 15 | . . . . 5
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌))) | 
| 36 | 30, 35 | jctird 526 | . . . 4
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌))))) | 
| 37 | 1, 3 | latmcl 18486 | . . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) | 
| 38 | 7, 37 | syl3an1 1163 | . . . . 5
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) | 
| 39 | 1, 8 | latasymb 18488 | . . . . 5
⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌))) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌))) | 
| 40 | 13, 27, 38, 39 | syl3anc 1372 | . . . 4
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌))) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌))) | 
| 41 | 36, 40 | sylibd 239 | . . 3
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌))) | 
| 42 | 12, 41 | impbid 212 | . 2
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌)) | 
| 43 | 6, 42 | bitrd 279 | 1
⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌)) |