Step |
Hyp |
Ref |
Expression |
1 |
|
hlhgt4.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
hlhgt4.s |
⊢ < = ( lt ‘ 𝐾 ) |
3 |
|
hlhgt4.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
hlhgt4.u |
⊢ 1 = ( 1. ‘ 𝐾 ) |
5 |
1 2 3 4
|
hlhgt4 |
⊢ ( 𝐾 ∈ HL → ∃ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ∃ 𝑧 ∈ 𝐵 ( ( 0 < 𝑦 ∧ 𝑦 < 𝑥 ) ∧ ( 𝑥 < 𝑧 ∧ 𝑧 < 1 ) ) ) |
6 |
|
hlpos |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Poset ) |
7 |
6
|
ad3antrrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → 𝐾 ∈ Poset ) |
8 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
9 |
8
|
ad3antrrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → 𝐾 ∈ OP ) |
10 |
1 3
|
op0cl |
⊢ ( 𝐾 ∈ OP → 0 ∈ 𝐵 ) |
11 |
9 10
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → 0 ∈ 𝐵 ) |
12 |
|
simpllr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
13 |
|
simplr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
14 |
1 2
|
plttr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( ( 0 < 𝑦 ∧ 𝑦 < 𝑥 ) → 0 < 𝑥 ) ) |
15 |
7 11 12 13 14
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → ( ( 0 < 𝑦 ∧ 𝑦 < 𝑥 ) → 0 < 𝑥 ) ) |
16 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → 𝑧 ∈ 𝐵 ) |
17 |
1 4
|
op1cl |
⊢ ( 𝐾 ∈ OP → 1 ∈ 𝐵 ) |
18 |
9 17
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → 1 ∈ 𝐵 ) |
19 |
1 2
|
plttr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 1 ∈ 𝐵 ) ) → ( ( 𝑥 < 𝑧 ∧ 𝑧 < 1 ) → 𝑥 < 1 ) ) |
20 |
7 13 16 18 19
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑥 < 𝑧 ∧ 𝑧 < 1 ) → 𝑥 < 1 ) ) |
21 |
15 20
|
anim12d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → ( ( ( 0 < 𝑦 ∧ 𝑦 < 𝑥 ) ∧ ( 𝑥 < 𝑧 ∧ 𝑧 < 1 ) ) → ( 0 < 𝑥 ∧ 𝑥 < 1 ) ) ) |
22 |
21
|
rexlimdva |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → ( ∃ 𝑧 ∈ 𝐵 ( ( 0 < 𝑦 ∧ 𝑦 < 𝑥 ) ∧ ( 𝑥 < 𝑧 ∧ 𝑧 < 1 ) ) → ( 0 < 𝑥 ∧ 𝑥 < 1 ) ) ) |
23 |
22
|
reximdva |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 ∃ 𝑧 ∈ 𝐵 ( ( 0 < 𝑦 ∧ 𝑦 < 𝑥 ) ∧ ( 𝑥 < 𝑧 ∧ 𝑧 < 1 ) ) → ∃ 𝑥 ∈ 𝐵 ( 0 < 𝑥 ∧ 𝑥 < 1 ) ) ) |
24 |
23
|
rexlimdva |
⊢ ( 𝐾 ∈ HL → ( ∃ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ∃ 𝑧 ∈ 𝐵 ( ( 0 < 𝑦 ∧ 𝑦 < 𝑥 ) ∧ ( 𝑥 < 𝑧 ∧ 𝑧 < 1 ) ) → ∃ 𝑥 ∈ 𝐵 ( 0 < 𝑥 ∧ 𝑥 < 1 ) ) ) |
25 |
5 24
|
mpd |
⊢ ( 𝐾 ∈ HL → ∃ 𝑥 ∈ 𝐵 ( 0 < 𝑥 ∧ 𝑥 < 1 ) ) |