Metamath Proof Explorer

Theorem atcv1

Description: Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion atcv1 
|- ( ( ( A e. CH /\ B e. HAtoms /\ C e. HAtoms ) /\ A  ( A = 0H <-> B = C ) )

Proof

Step Hyp Ref Expression
1 breq1 
 |-  ( A = 0H -> ( A  0H
2 atcv0eq 
 |-  ( ( B e. HAtoms /\ C e. HAtoms ) -> ( 0H  B = C ) )
3 1 2 sylan9bbr 
 |-  ( ( ( B e. HAtoms /\ C e. HAtoms ) /\ A = 0H ) -> ( A  B = C ) )
4 3 biimpd 
 |-  ( ( ( B e. HAtoms /\ C e. HAtoms ) /\ A = 0H ) -> ( A  B = C ) )
5 4 ex 
 |-  ( ( B e. HAtoms /\ C e. HAtoms ) -> ( A = 0H -> ( A  B = C ) ) )
6 5 com23 
 |-  ( ( B e. HAtoms /\ C e. HAtoms ) -> ( A  ( A = 0H -> B = C ) ) )
7 6 3adant1 
 |-  ( ( A e. CH /\ B e. HAtoms /\ C e. HAtoms ) -> ( A  ( A = 0H -> B = C ) ) )
8 7 imp 
 |-  ( ( ( A e. CH /\ B e. HAtoms /\ C e. HAtoms ) /\ A  ( A = 0H -> B = C ) )
9 oveq1 
 |-  ( B = C -> ( B vH C ) = ( C vH C ) )
10 atelch 
 |-  ( C e. HAtoms -> C e. CH )
11 chjidm 
 |-  ( C e. CH -> ( C vH C ) = C )
12 10 11 syl 
 |-  ( C e. HAtoms -> ( C vH C ) = C )
13 9 12 sylan9eq 
 |-  ( ( B = C /\ C e. HAtoms ) -> ( B vH C ) = C )
14 13 eqcomd 
 |-  ( ( B = C /\ C e. HAtoms ) -> C = ( B vH C ) )
15 14 eleq1d 
 |-  ( ( B = C /\ C e. HAtoms ) -> ( C e. HAtoms <-> ( B vH C ) e. HAtoms ) )
16 15 ex 
 |-  ( B = C -> ( C e. HAtoms -> ( C e. HAtoms <-> ( B vH C ) e. HAtoms ) ) )
17 16 ibd 
 |-  ( B = C -> ( C e. HAtoms -> ( B vH C ) e. HAtoms ) )
18 17 impcom 
 |-  ( ( C e. HAtoms /\ B = C ) -> ( B vH C ) e. HAtoms )
19 atcveq0 
 |-  ( ( A e. CH /\ ( B vH C ) e. HAtoms ) -> ( A  A = 0H ) )
20 18 19 sylan2 
 |-  ( ( A e. CH /\ ( C e. HAtoms /\ B = C ) ) -> ( A  A = 0H ) )
21 20 biimpd 
 |-  ( ( A e. CH /\ ( C e. HAtoms /\ B = C ) ) -> ( A  A = 0H ) )
22 21 exp32 
 |-  ( A e. CH -> ( C e. HAtoms -> ( B = C -> ( A  A = 0H ) ) ) )
23 22 com34 
 |-  ( A e. CH -> ( C e. HAtoms -> ( A  ( B = C -> A = 0H ) ) ) )
24 23 imp 
 |-  ( ( A e. CH /\ C e. HAtoms ) -> ( A  ( B = C -> A = 0H ) ) )
25 24 3adant2 
 |-  ( ( A e. CH /\ B e. HAtoms /\ C e. HAtoms ) -> ( A  ( B = C -> A = 0H ) ) )
26 25 imp 
 |-  ( ( ( A e. CH /\ B e. HAtoms /\ C e. HAtoms ) /\ A  ( B = C -> A = 0H ) )
27 8 26 impbid 
 |-  ( ( ( A e. CH /\ B e. HAtoms /\ C e. HAtoms ) /\ A  ( A = 0H <-> B = C ) )