Metamath Proof Explorer


Theorem fh4i

Description: Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005) (New usage is discouraged.)

Ref Expression
Hypotheses fh1.1 𝐴C
fh1.2 𝐵C
fh1.3 𝐶C
fh1.4 𝐴 𝐶 𝐵
fh1.5 𝐴 𝐶 𝐶
Assertion fh4i ( 𝐵 ( 𝐴𝐶 ) ) = ( ( 𝐵 𝐴 ) ∩ ( 𝐵 𝐶 ) )

Proof

Step Hyp Ref Expression
1 fh1.1 𝐴C
2 fh1.2 𝐵C
3 fh1.3 𝐶C
4 fh1.4 𝐴 𝐶 𝐵
5 fh1.5 𝐴 𝐶 𝐶
6 1 choccli ( ⊥ ‘ 𝐴 ) ∈ C
7 2 choccli ( ⊥ ‘ 𝐵 ) ∈ C
8 3 choccli ( ⊥ ‘ 𝐶 ) ∈ C
9 1 2 4 cmcm3ii ( ⊥ ‘ 𝐴 ) 𝐶 𝐵
10 6 2 9 cmcm2ii ( ⊥ ‘ 𝐴 ) 𝐶 ( ⊥ ‘ 𝐵 )
11 1 3 5 cmcm3ii ( ⊥ ‘ 𝐴 ) 𝐶 𝐶
12 6 3 11 cmcm2ii ( ⊥ ‘ 𝐴 ) 𝐶 ( ⊥ ‘ 𝐶 )
13 6 7 8 10 12 fh2i ( ( ⊥ ‘ 𝐵 ) ∩ ( ( ⊥ ‘ 𝐴 ) ∨ ( ⊥ ‘ 𝐶 ) ) ) = ( ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐴 ) ) ∨ ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐶 ) ) )
14 1 3 chdmm1i ( ⊥ ‘ ( 𝐴𝐶 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ ( ⊥ ‘ 𝐶 ) )
15 14 ineq2i ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ ( 𝐴𝐶 ) ) ) = ( ( ⊥ ‘ 𝐵 ) ∩ ( ( ⊥ ‘ 𝐴 ) ∨ ( ⊥ ‘ 𝐶 ) ) )
16 2 1 chdmj1i ( ⊥ ‘ ( 𝐵 𝐴 ) ) = ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐴 ) )
17 2 3 chdmj1i ( ⊥ ‘ ( 𝐵 𝐶 ) ) = ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐶 ) )
18 16 17 oveq12i ( ( ⊥ ‘ ( 𝐵 𝐴 ) ) ∨ ( ⊥ ‘ ( 𝐵 𝐶 ) ) ) = ( ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐴 ) ) ∨ ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐶 ) ) )
19 13 15 18 3eqtr4ri ( ( ⊥ ‘ ( 𝐵 𝐴 ) ) ∨ ( ⊥ ‘ ( 𝐵 𝐶 ) ) ) = ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ ( 𝐴𝐶 ) ) )
20 2 1 chjcli ( 𝐵 𝐴 ) ∈ C
21 2 3 chjcli ( 𝐵 𝐶 ) ∈ C
22 20 21 chdmm1i ( ⊥ ‘ ( ( 𝐵 𝐴 ) ∩ ( 𝐵 𝐶 ) ) ) = ( ( ⊥ ‘ ( 𝐵 𝐴 ) ) ∨ ( ⊥ ‘ ( 𝐵 𝐶 ) ) )
23 1 3 chincli ( 𝐴𝐶 ) ∈ C
24 2 23 chdmj1i ( ⊥ ‘ ( 𝐵 ( 𝐴𝐶 ) ) ) = ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ ( 𝐴𝐶 ) ) )
25 19 22 24 3eqtr4i ( ⊥ ‘ ( ( 𝐵 𝐴 ) ∩ ( 𝐵 𝐶 ) ) ) = ( ⊥ ‘ ( 𝐵 ( 𝐴𝐶 ) ) )
26 2 23 chjcli ( 𝐵 ( 𝐴𝐶 ) ) ∈ C
27 20 21 chincli ( ( 𝐵 𝐴 ) ∩ ( 𝐵 𝐶 ) ) ∈ C
28 26 27 chcon3i ( ( 𝐵 ( 𝐴𝐶 ) ) = ( ( 𝐵 𝐴 ) ∩ ( 𝐵 𝐶 ) ) ↔ ( ⊥ ‘ ( ( 𝐵 𝐴 ) ∩ ( 𝐵 𝐶 ) ) ) = ( ⊥ ‘ ( 𝐵 ( 𝐴𝐶 ) ) ) )
29 25 28 mpbir ( 𝐵 ( 𝐴𝐶 ) ) = ( ( 𝐵 𝐴 ) ∩ ( 𝐵 𝐶 ) )