Step |
Hyp |
Ref |
Expression |
1 |
|
atcvrj0.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
atcvrj0.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
atcvrj0.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
atcvrj0.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
5 |
|
atcvrj0.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
breq1 |
⊢ ( 𝑋 = 0 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 0 𝐶 ( 𝑃 ∨ 𝑄 ) ) ) |
7 |
6
|
biimpd |
⊢ ( 𝑋 = 0 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 0 𝐶 ( 𝑃 ∨ 𝑄 ) ) ) |
8 |
7
|
adantl |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 = 0 ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 0 𝐶 ( 𝑃 ∨ 𝑄 ) ) ) |
9 |
2 3 4 5
|
atcvr0eq |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 0 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 = 𝑄 ) ) |
10 |
9
|
3adant3r1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 0 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 = 𝑄 ) ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 = 0 ) → ( 0 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 = 𝑄 ) ) |
12 |
8 11
|
sylibd |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 = 0 ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 𝑃 = 𝑄 ) ) |
13 |
12
|
ex |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 = 0 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 𝑃 = 𝑄 ) ) ) |
14 |
13
|
com23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → ( 𝑋 = 0 → 𝑃 = 𝑄 ) ) ) |
15 |
14
|
3impia |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 = 0 → 𝑃 = 𝑄 ) ) |
16 |
|
oveq1 |
⊢ ( 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑄 ) ) |
17 |
16
|
breq2d |
⊢ ( 𝑃 = 𝑄 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 𝑋 𝐶 ( 𝑄 ∨ 𝑄 ) ) ) |
18 |
17
|
biimpac |
⊢ ( ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 = 𝑄 ) → 𝑋 𝐶 ( 𝑄 ∨ 𝑄 ) ) |
19 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 ) |
20 |
2 5
|
hlatjidm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑄 ) = 𝑄 ) |
21 |
19 20
|
syldan |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑄 ∨ 𝑄 ) = 𝑄 ) |
22 |
21
|
breq2d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑄 ∨ 𝑄 ) ↔ 𝑋 𝐶 𝑄 ) ) |
23 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
24 |
23
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ AtLat ) |
25 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑋 ∈ 𝐵 ) |
26 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
27 |
1 26 3 4 5
|
atcvreq0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑋 𝐶 𝑄 ↔ 𝑋 = 0 ) ) |
28 |
24 25 19 27
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 𝑄 ↔ 𝑋 = 0 ) ) |
29 |
28
|
biimpd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 𝑄 → 𝑋 = 0 ) ) |
30 |
22 29
|
sylbid |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑄 ∨ 𝑄 ) → 𝑋 = 0 ) ) |
31 |
18 30
|
syl5 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 = 𝑄 ) → 𝑋 = 0 ) ) |
32 |
31
|
expd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → ( 𝑃 = 𝑄 → 𝑋 = 0 ) ) ) |
33 |
32
|
3impia |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 = 𝑄 → 𝑋 = 0 ) ) |
34 |
15 33
|
impbid |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 = 0 ↔ 𝑃 = 𝑄 ) ) |