Step |
Hyp |
Ref |
Expression |
1 |
|
4that.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
4that.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
4that.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
4that.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
simp21l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 = ( 0. ‘ 𝐾 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑃 ∈ 𝐴 ) |
6 |
|
simp21r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 = ( 0. ‘ 𝐾 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ¬ 𝑃 ≤ 𝑊 ) |
7 |
|
simp23 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 = ( 0. ‘ 𝐾 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑆 = ( 0. ‘ 𝐾 ) ) |
8 |
7
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 = ( 0. ‘ 𝐾 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑆 ∨ 𝑃 ) = ( ( 0. ‘ 𝐾 ) ∨ 𝑃 ) ) |
9 |
|
simp32 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 = ( 0. ‘ 𝐾 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑇 = ( 0. ‘ 𝐾 ) ) |
10 |
9
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 = ( 0. ‘ 𝐾 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑇 ∨ 𝑃 ) = ( ( 0. ‘ 𝐾 ) ∨ 𝑃 ) ) |
11 |
8 10
|
eqtr4d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 = ( 0. ‘ 𝐾 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑆 ∨ 𝑃 ) = ( 𝑇 ∨ 𝑃 ) ) |
12 |
|
breq1 |
⊢ ( 𝑧 = 𝑃 → ( 𝑧 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊 ) ) |
13 |
12
|
notbid |
⊢ ( 𝑧 = 𝑃 → ( ¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊 ) ) |
14 |
|
oveq2 |
⊢ ( 𝑧 = 𝑃 → ( 𝑆 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑃 ) ) |
15 |
|
oveq2 |
⊢ ( 𝑧 = 𝑃 → ( 𝑇 ∨ 𝑧 ) = ( 𝑇 ∨ 𝑃 ) ) |
16 |
14 15
|
eqeq12d |
⊢ ( 𝑧 = 𝑃 → ( ( 𝑆 ∨ 𝑧 ) = ( 𝑇 ∨ 𝑧 ) ↔ ( 𝑆 ∨ 𝑃 ) = ( 𝑇 ∨ 𝑃 ) ) ) |
17 |
13 16
|
anbi12d |
⊢ ( 𝑧 = 𝑃 → ( ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑆 ∨ 𝑧 ) = ( 𝑇 ∨ 𝑧 ) ) ↔ ( ¬ 𝑃 ≤ 𝑊 ∧ ( 𝑆 ∨ 𝑃 ) = ( 𝑇 ∨ 𝑃 ) ) ) ) |
18 |
17
|
rspcev |
⊢ ( ( 𝑃 ∈ 𝐴 ∧ ( ¬ 𝑃 ≤ 𝑊 ∧ ( 𝑆 ∨ 𝑃 ) = ( 𝑇 ∨ 𝑃 ) ) ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑆 ∨ 𝑧 ) = ( 𝑇 ∨ 𝑧 ) ) ) |
19 |
5 6 11 18
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 = ( 0. ‘ 𝐾 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑆 ∨ 𝑧 ) = ( 𝑇 ∨ 𝑧 ) ) ) |