Description: Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | shincl.1 | |- A e. SH |
|
| shincl.2 | |- B e. SH |
||
| shless.1 | |- C e. SH |
||
| Assertion | shlej1i | |- ( A C_ B -> ( A vH C ) C_ ( B vH C ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | shincl.1 | |- A e. SH |
|
| 2 | shincl.2 | |- B e. SH |
|
| 3 | shless.1 | |- C e. SH |
|
| 4 | shlej1 | |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( A vH C ) C_ ( B vH C ) ) |
|
| 5 | 4 | ex | |- ( ( A e. SH /\ B e. SH /\ C e. SH ) -> ( A C_ B -> ( A vH C ) C_ ( B vH C ) ) ) |
| 6 | 1 2 3 5 | mp3an | |- ( A C_ B -> ( A vH C ) C_ ( B vH C ) ) |