Metamath Proof Explorer


Theorem chlej1i

Description: Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses ch0le.1 𝐴C
chjcl.2 𝐵C
chlub.1 𝐶C
Assertion chlej1i ( 𝐴𝐵 → ( 𝐴 𝐶 ) ⊆ ( 𝐵 𝐶 ) )

Proof

Step Hyp Ref Expression
1 ch0le.1 𝐴C
2 chjcl.2 𝐵C
3 chlub.1 𝐶C
4 1 chshii 𝐴S
5 2 chshii 𝐵S
6 3 chshii 𝐶S
7 4 5 6 shlej1i ( 𝐴𝐵 → ( 𝐴 𝐶 ) ⊆ ( 𝐵 𝐶 ) )