Step |
Hyp |
Ref |
Expression |
1 |
|
1cvratlt.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
1cvratlt.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
1cvratlt.s |
⊢ < = ( lt ‘ 𝐾 ) |
4 |
|
1cvratlt.u |
⊢ 1 = ( 1. ‘ 𝐾 ) |
5 |
|
1cvratlt.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
6 |
|
1cvratlt.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) → 𝐾 ∈ HL ) |
8 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
9 |
|
simprl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) → 𝑋 𝐶 1 ) |
10 |
1 3 4 5 6
|
1cvratex |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 𝐶 1 ) → ∃ 𝑞 ∈ 𝐴 𝑞 < 𝑋 ) |
11 |
7 8 9 10
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) → ∃ 𝑞 ∈ 𝐴 𝑞 < 𝑋 ) |
12 |
|
simp1l1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋 ) → 𝐾 ∈ HL ) |
13 |
|
simp1l2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋 ) → 𝑃 ∈ 𝐴 ) |
14 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋 ) → 𝑞 ∈ 𝐴 ) |
15 |
|
simp1l3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋 ) → 𝑋 ∈ 𝐵 ) |
16 |
|
simp1rr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋 ) → 𝑃 ≤ 𝑋 ) |
17 |
|
simp3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋 ) → 𝑞 < 𝑋 ) |
18 |
1 2 3 6
|
atlelt |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑞 < 𝑋 ) ) → 𝑃 < 𝑋 ) |
19 |
12 13 14 15 16 17 18
|
syl132anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋 ) → 𝑃 < 𝑋 ) |
20 |
19
|
rexlimdv3a |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) → ( ∃ 𝑞 ∈ 𝐴 𝑞 < 𝑋 → 𝑃 < 𝑋 ) ) |
21 |
11 20
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑋 𝐶 1 ∧ 𝑃 ≤ 𝑋 ) ) → 𝑃 < 𝑋 ) |