Metamath Proof Explorer


Theorem dihmeetlem14N

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 )
Assertion dihmeetlem14N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵𝑝𝐵 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ( ( 𝐼𝑟 ) ∩ ( 𝐼𝑝 ) ) ) = ( ( 𝐼𝑌 ) ∩ ( 𝐼𝑝 ) ) )

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 1 2 3 4 5 6 7 8 9 dihmeetlem12N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵𝑝𝐵 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑌 𝑝 ) ) ( ( 𝐼𝑟 ) ∩ ( 𝐼𝑝 ) ) ) = ( ( 𝐼𝑌 ) ∩ ( 𝐼𝑝 ) ) )