Metamath Proof Explorer


Theorem dihmeetlem17N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem14.b 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem14.l = ( le ‘ 𝐾 )
dihmeetlem14.h 𝐻 = ( LHyp ‘ 𝐾 )
dihmeetlem14.j = ( join ‘ 𝐾 )
dihmeetlem14.m = ( meet ‘ 𝐾 )
dihmeetlem14.a 𝐴 = ( Atoms ‘ 𝐾 )
dihmeetlem14.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihmeetlem14.s = ( LSSum ‘ 𝑈 )
dihmeetlem14.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihmeetlem17.o 0 = ( 0. ‘ 𝐾 )
Assertion dihmeetlem17N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( 𝑌 𝑝 ) = 0 )

Proof

Step Hyp Ref Expression
1 dihmeetlem14.b 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem14.l = ( le ‘ 𝐾 )
3 dihmeetlem14.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dihmeetlem14.j = ( join ‘ 𝐾 )
5 dihmeetlem14.m = ( meet ‘ 𝐾 )
6 dihmeetlem14.a 𝐴 = ( Atoms ‘ 𝐾 )
7 dihmeetlem14.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihmeetlem14.s = ( LSSum ‘ 𝑈 )
9 dihmeetlem14.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 dihmeetlem17.o 0 = ( 0. ‘ 𝐾 )
11 simpl1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → 𝐾 ∈ HL )
12 11 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → 𝐾 ∈ Lat )
13 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → 𝑝𝐴 )
14 1 6 atbase ( 𝑝𝐴𝑝𝐵 )
15 13 14 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → 𝑝𝐵 )
16 simpr1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → 𝑌𝐵 )
17 1 5 latmcom ( ( 𝐾 ∈ Lat ∧ 𝑝𝐵𝑌𝐵 ) → ( 𝑝 𝑌 ) = ( 𝑌 𝑝 ) )
18 12 15 16 17 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( 𝑝 𝑌 ) = ( 𝑌 𝑝 ) )
19 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
20 simpl2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) )
21 simpl3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) )
22 simpr2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( 𝑋 𝑌 ) 𝑊 )
23 simpr3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → 𝑝 𝑋 )
24 1 2 4 5 6 3 lhpmcvr4N ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ¬ 𝑝 𝑌 )
25 19 20 21 16 22 23 24 syl123anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ¬ 𝑝 𝑌 )
26 hlatl ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
27 11 26 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → 𝐾 ∈ AtLat )
28 1 2 5 10 6 atnle ( ( 𝐾 ∈ AtLat ∧ 𝑝𝐴𝑌𝐵 ) → ( ¬ 𝑝 𝑌 ↔ ( 𝑝 𝑌 ) = 0 ) )
29 27 13 16 28 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( ¬ 𝑝 𝑌 ↔ ( 𝑝 𝑌 ) = 0 ) )
30 25 29 mpbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( 𝑝 𝑌 ) = 0 )
31 18 30 eqtr3d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑋 𝑌 ) 𝑊𝑝 𝑋 ) ) → ( 𝑌 𝑝 ) = 0 )