Metamath Proof Explorer


Theorem dihglblem5aN

Description: A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihglblem5a.b 𝐵 = ( Base ‘ 𝐾 )
dihglblem5a.m = ( meet ‘ 𝐾 )
dihglblem5a.h 𝐻 = ( LHyp ‘ 𝐾 )
dihglblem5a.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihglblem5aN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 dihglblem5a.b 𝐵 = ( Base ‘ 𝐾 )
2 dihglblem5a.m = ( meet ‘ 𝐾 )
3 dihglblem5a.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dihglblem5a.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → 𝑋 ( le ‘ 𝐾 ) 𝑊 )
6 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
7 6 ad3antrrr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → 𝐾 ∈ Lat )
8 simplr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → 𝑋𝐵 )
9 1 3 lhpbase ( 𝑊𝐻𝑊𝐵 )
10 9 ad3antlr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → 𝑊𝐵 )
11 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
12 1 11 2 latleeqm1 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑊 ↔ ( 𝑋 𝑊 ) = 𝑋 ) )
13 7 8 10 12 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑊 ↔ ( 𝑋 𝑊 ) = 𝑋 ) )
14 5 13 mpbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝑋 𝑊 ) = 𝑋 )
15 14 fveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) = ( 𝐼𝑋 ) )
16 simpll ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 1 11 3 4 dihord ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑊𝐵 ) → ( ( 𝐼𝑋 ) ⊆ ( 𝐼𝑊 ) ↔ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) )
18 16 8 10 17 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝐼𝑋 ) ⊆ ( 𝐼𝑊 ) ↔ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) )
19 5 18 mpbird ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑊 ) )
20 df-ss ( ( 𝐼𝑋 ) ⊆ ( 𝐼𝑊 ) ↔ ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑊 ) ) = ( 𝐼𝑋 ) )
21 19 20 sylib ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑊 ) ) = ( 𝐼𝑋 ) )
22 15 21 eqtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑊 ) ) )
23 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
24 eqid ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
25 eqid ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
26 eqid ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
27 eqid ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
28 eqid ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
29 eqid ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 ) = ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 )
30 eqid ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
31 1 2 3 4 11 23 24 25 26 27 28 29 30 dihglblem5apreN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑊 ) ) )
32 31 anassrs ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) ∧ ¬ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑊 ) ) )
33 22 32 pm2.61dan ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵 ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) = ( ( 𝐼𝑋 ) ∩ ( 𝐼𝑊 ) ) )