Step |
Hyp |
Ref |
Expression |
1 |
|
cvrat2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cvrat2.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cvrat2.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
4 |
|
cvrat2.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
6 |
1 2 5 3 4
|
atcvrj0 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 = ( 0. ‘ 𝐾 ) ↔ 𝑃 = 𝑄 ) ) |
7 |
6
|
3expa |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 = ( 0. ‘ 𝐾 ) ↔ 𝑃 = 𝑄 ) ) |
8 |
7
|
necon3bid |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 ≠ ( 0. ‘ 𝐾 ) ↔ 𝑃 ≠ 𝑄 ) ) |
9 |
|
simpl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ HL ) |
10 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑋 ∈ 𝐵 ) |
11 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
12 |
11
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ Lat ) |
13 |
|
simpr2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 ) |
14 |
1 4
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐵 ) |
16 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 ) |
17 |
1 4
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
18 |
16 17
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐵 ) |
19 |
1 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) |
20 |
12 15 18 19
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) |
21 |
|
eqid |
⊢ ( lt ‘ 𝐾 ) = ( lt ‘ 𝐾 ) |
22 |
1 21 3
|
cvrlt |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → 𝑋 ( lt ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) |
23 |
22
|
ex |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 𝑋 ( lt ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ) |
24 |
9 10 20 23
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 𝑋 ( lt ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ) |
25 |
1 21 2 5 4
|
cvrat |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 ≠ ( 0. ‘ 𝐾 ) ∧ 𝑋 ( lt ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) → 𝑋 ∈ 𝐴 ) ) |
26 |
25
|
expcomd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 ( lt ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) → ( 𝑋 ≠ ( 0. ‘ 𝐾 ) → 𝑋 ∈ 𝐴 ) ) ) |
27 |
24 26
|
syld |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → ( 𝑋 ≠ ( 0. ‘ 𝐾 ) → 𝑋 ∈ 𝐴 ) ) ) |
28 |
27
|
imp |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 ≠ ( 0. ‘ 𝐾 ) → 𝑋 ∈ 𝐴 ) ) |
29 |
8 28
|
sylbird |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ≠ 𝑄 → 𝑋 ∈ 𝐴 ) ) |
30 |
29
|
ex |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → ( 𝑃 ≠ 𝑄 → 𝑋 ∈ 𝐴 ) ) ) |
31 |
30
|
com23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ≠ 𝑄 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 𝑋 ∈ 𝐴 ) ) ) |
32 |
31
|
impd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → 𝑋 ∈ 𝐴 ) ) |
33 |
32
|
3impia |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) ) → 𝑋 ∈ 𝐴 ) |