Metamath Proof Explorer


Theorem isline4N

Description: Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012) (New usage is discouraged.)

Ref Expression
Hypotheses isline4.b 𝐵 = ( Base ‘ 𝐾 )
isline4.c 𝐶 = ( ⋖ ‘ 𝐾 )
isline4.a 𝐴 = ( Atoms ‘ 𝐾 )
isline4.n 𝑁 = ( Lines ‘ 𝐾 )
isline4.m 𝑀 = ( pmap ‘ 𝐾 )
Assertion isline4N ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) → ( ( 𝑀𝑋 ) ∈ 𝑁 ↔ ∃ 𝑝𝐴 𝑝 𝐶 𝑋 ) )

Proof

Step Hyp Ref Expression
1 isline4.b 𝐵 = ( Base ‘ 𝐾 )
2 isline4.c 𝐶 = ( ⋖ ‘ 𝐾 )
3 isline4.a 𝐴 = ( Atoms ‘ 𝐾 )
4 isline4.n 𝑁 = ( Lines ‘ 𝐾 )
5 isline4.m 𝑀 = ( pmap ‘ 𝐾 )
6 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
7 1 6 3 4 5 isline3 ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) → ( ( 𝑀𝑋 ) ∈ 𝑁 ↔ ∃ 𝑝𝐴𝑞𝐴 ( 𝑝𝑞𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
8 simpll ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) → 𝐾 ∈ HL )
9 1 3 atbase ( 𝑝𝐴𝑝𝐵 )
10 9 adantl ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) → 𝑝𝐵 )
11 simplr ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) → 𝑋𝐵 )
12 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
13 1 12 6 2 3 cvrval3 ( ( 𝐾 ∈ HL ∧ 𝑝𝐵𝑋𝐵 ) → ( 𝑝 𝐶 𝑋 ↔ ∃ 𝑞𝐴 ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ∧ ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ) ) )
14 8 10 11 13 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) → ( 𝑝 𝐶 𝑋 ↔ ∃ 𝑞𝐴 ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ∧ ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ) ) )
15 hlatl ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
16 15 ad3antrrr ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) ∧ 𝑞𝐴 ) → 𝐾 ∈ AtLat )
17 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) ∧ 𝑞𝐴 ) → 𝑞𝐴 )
18 simplr ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) ∧ 𝑞𝐴 ) → 𝑝𝐴 )
19 12 3 atncmp ( ( 𝐾 ∈ AtLat ∧ 𝑞𝐴𝑝𝐴 ) → ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝𝑞𝑝 ) )
20 16 17 18 19 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) ∧ 𝑞𝐴 ) → ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝𝑞𝑝 ) )
21 necom ( 𝑞𝑝𝑝𝑞 )
22 20 21 bitrdi ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) ∧ 𝑞𝐴 ) → ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝𝑝𝑞 ) )
23 eqcom ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) )
24 23 a1i ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) ∧ 𝑞𝐴 ) → ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) )
25 22 24 anbi12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) ∧ 𝑞𝐴 ) → ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ∧ ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ) ↔ ( 𝑝𝑞𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
26 25 rexbidva ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) → ( ∃ 𝑞𝐴 ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ∧ ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ) ↔ ∃ 𝑞𝐴 ( 𝑝𝑞𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
27 14 26 bitrd ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) ∧ 𝑝𝐴 ) → ( 𝑝 𝐶 𝑋 ↔ ∃ 𝑞𝐴 ( 𝑝𝑞𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
28 27 rexbidva ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) → ( ∃ 𝑝𝐴 𝑝 𝐶 𝑋 ↔ ∃ 𝑝𝐴𝑞𝐴 ( 𝑝𝑞𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
29 7 28 bitr4d ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ) → ( ( 𝑀𝑋 ) ∈ 𝑁 ↔ ∃ 𝑝𝐴 𝑝 𝐶 𝑋 ) )