Metamath Proof Explorer


Theorem pjocini

Description: Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001) (New usage is discouraged.)

Ref Expression
Hypotheses pjocin.1 GC
pjocin.2 HC
Assertion pjocini AGHprojGAGH

Proof

Step Hyp Ref Expression
1 pjocin.1 GC
2 pjocin.2 HC
3 1 2 chincli GHC
4 3 choccli GHC
5 4 cheli AGHA
6 pjpo GCAprojGA=A-projGA
7 1 5 6 sylancr AGHprojGA=A-projGA
8 inss1 GHG
9 3 1 chsscon3i GHGGGH
10 8 9 mpbi GGH
11 1 choccli GC
12 11 pjcli AprojGAG
13 5 12 syl AGHprojGAG
14 10 13 sselid AGHprojGAGH
15 4 chshii GHS
16 shsubcl GHSAGHprojGAGHA-projGAGH
17 15 16 mp3an1 AGHprojGAGHA-projGAGH
18 14 17 mpdan AGHA-projGAGH
19 7 18 eqeltrd AGHprojGAGH