| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nonconne 2952 | . 2
⊢  ¬
(𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋) | 
| 2 |  | simpl1 1192 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝐾 ∈ HL) | 
| 3 |  | simpl2 1193 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝑋 ∈ 𝐶) | 
| 4 |  | eqid 2737 | . . . . . . . 8
⊢
(Atoms‘𝐾) =
(Atoms‘𝐾) | 
| 5 |  | osumcl.c | . . . . . . . 8
⊢ 𝐶 = (PSubCl‘𝐾) | 
| 6 | 4, 5 | psubclssatN 39943 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾)) | 
| 7 | 2, 3, 6 | syl2anc 584 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ (Atoms‘𝐾)) | 
| 8 |  | simpl3 1194 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝑌 ∈ 𝐶) | 
| 9 | 4, 5 | psubclssatN 39943 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → 𝑌 ⊆ (Atoms‘𝐾)) | 
| 10 | 2, 8, 9 | syl2anc 584 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝑌 ⊆ (Atoms‘𝐾)) | 
| 11 |  | osumcl.p | . . . . . . 7
⊢  + =
(+𝑃‘𝐾) | 
| 12 | 4, 11 | paddssat 39816 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) | 
| 13 | 2, 7, 10, 12 | syl3anc 1373 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) | 
| 14 |  | osumcl.o | . . . . . 6
⊢  ⊥ =
(⊥𝑃‘𝐾) | 
| 15 | 4, 14 | 2polssN 39917 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) | 
| 16 | 2, 13, 15 | syl2anc 584 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) | 
| 17 |  | df-pss 3971 | . . . . . . 7
⊢ ((𝑋 + 𝑌) ⊊ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ↔ ((𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ∧ (𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))))) | 
| 18 |  | pssnel 4471 | . . . . . . 7
⊢ ((𝑋 + 𝑌) ⊊ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) → ∃𝑝(𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌))) | 
| 19 | 17, 18 | sylbir 235 | . . . . . 6
⊢ (((𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ∧ (𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) → ∃𝑝(𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌))) | 
| 20 |  | df-rex 3071 | . . . . . 6
⊢
(∃𝑝 ∈ (
⊥
‘( ⊥ ‘(𝑋 + 𝑌))) ¬ 𝑝 ∈ (𝑋 + 𝑌) ↔ ∃𝑝(𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌))) | 
| 21 | 19, 20 | sylibr 234 | . . . . 5
⊢ (((𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ∧ (𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) → ∃𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ¬ 𝑝 ∈ (𝑋 + 𝑌)) | 
| 22 |  | eqid 2737 | . . . . . . . . . . 11
⊢
(le‘𝐾) =
(le‘𝐾) | 
| 23 |  | eqid 2737 | . . . . . . . . . . 11
⊢
(join‘𝐾) =
(join‘𝐾) | 
| 24 |  | eqid 2737 | . . . . . . . . . . 11
⊢ (𝑋 + {𝑝}) = (𝑋 + {𝑝}) | 
| 25 |  | eqid 2737 | . . . . . . . . . . 11
⊢ ( ⊥
‘( ⊥ ‘(𝑋 + 𝑌))) = ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) | 
| 26 | 22, 23, 4, 11, 14, 5, 24, 25 | osumcllem9N 39966 | . . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) = 𝑋) | 
| 27 |  | simp11 1204 | . . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝐾 ∈ HL) | 
| 28 |  | simp12 1205 | . . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ∈ 𝐶) | 
| 29 | 27, 28, 6 | syl2anc 584 | . . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ (Atoms‘𝐾)) | 
| 30 |  | simp13 1206 | . . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌 ∈ 𝐶) | 
| 31 | 27, 30, 9 | syl2anc 584 | . . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌 ⊆ (Atoms‘𝐾)) | 
| 32 | 13 | 3adantr3 1172 | . . . . . . . . . . . . . . 15
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))))) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) | 
| 33 | 32 | 3adant3 1133 | . . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) | 
| 34 | 4, 14 | polssatN 39910 | . . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ (Atoms‘𝐾)) | 
| 35 | 27, 33, 34 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ (Atoms‘𝐾)) | 
| 36 | 4, 14 | polssatN 39910 | . . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ ( ⊥
‘(𝑋 + 𝑌)) ⊆ (Atoms‘𝐾)) → ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ⊆ (Atoms‘𝐾)) | 
| 37 | 27, 35, 36 | syl2anc 584 | . . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ⊆ (Atoms‘𝐾)) | 
| 38 |  | simp23 1209 | . . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) | 
| 39 | 37, 38 | sseldd 3984 | . . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ (Atoms‘𝐾)) | 
| 40 |  | simp3 1139 | . . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ¬ 𝑝 ∈ (𝑋 + 𝑌)) | 
| 41 | 22, 23, 4, 11, 14, 5, 24, 25 | osumcllem10N 39967 | . . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ≠ 𝑋) | 
| 42 | 27, 29, 31, 39, 40, 41 | syl311anc 1386 | . . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ≠ 𝑋) | 
| 43 | 26, 42 | pm2.21ddne 3026 | . . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)) | 
| 44 | 43 | 3exp 1120 | . . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) → ((𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) → (¬ 𝑝 ∈ (𝑋 + 𝑌) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))) | 
| 45 | 44 | 3expd 1354 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) → (𝑋 ⊆ ( ⊥ ‘𝑌) → (𝑋 ≠ ∅ → (𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) → (¬ 𝑝 ∈ (𝑋 + 𝑌) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))))) | 
| 46 | 45 | imp32 418 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) → (¬ 𝑝 ∈ (𝑋 + 𝑌) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))) | 
| 47 | 46 | rexlimdv 3153 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (∃𝑝 ∈ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ¬ 𝑝 ∈ (𝑋 + 𝑌) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))) | 
| 48 | 21, 47 | syl5 34 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (((𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) ∧ (𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))) | 
| 49 | 16, 48 | mpand 695 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → ((𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))) | 
| 50 | 49 | necon1bd 2958 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋) → (𝑋 + 𝑌) = ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌))))) | 
| 51 | 1, 50 | mpi 20 | 1
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) = ( ⊥ ‘( ⊥
‘(𝑋 + 𝑌)))) |