Metamath Proof Explorer

Theorem padd4N

Description: Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses paddass.a 
|- A = ( Atoms ` K )
paddass.p 
|- .+ = ( +P ` K )
Assertion padd4N 
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Proof

Step Hyp Ref Expression
1 paddass.a 
 |-  A = ( Atoms ` K )
2 paddass.p 
 |-  .+ = ( +P ` K )
3 simp1 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> K e. HL )
4 simp2r 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> Y C_ A )
5 simp3l 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> Z C_ A )
6 simp3r 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> W C_ A )
7 1 2 padd12N 
 |-  ( ( K e. HL /\ ( Y C_ A /\ Z C_ A /\ W C_ A ) ) -> ( Y .+ ( Z .+ W ) ) = ( Z .+ ( Y .+ W ) ) )
8 3 4 5 6 7 syl13anc 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> ( Y .+ ( Z .+ W ) ) = ( Z .+ ( Y .+ W ) ) )
9 8 oveq2d 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> ( X .+ ( Y .+ ( Z .+ W ) ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
10 simp2l 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> X C_ A )
11 1 2 paddssat 
 |-  ( ( K e. HL /\ Z C_ A /\ W C_ A ) -> ( Z .+ W ) C_ A )
12 3 5 6 11 syl3anc 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> ( Z .+ W ) C_ A )
13 1 2 paddass 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ ( Z .+ W ) C_ A ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( X .+ ( Y .+ ( Z .+ W ) ) ) )
14 3 10 4 12 13 syl13anc 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( X .+ ( Y .+ ( Z .+ W ) ) ) )
15 1 2 paddssat 
 |-  ( ( K e. HL /\ Y C_ A /\ W C_ A ) -> ( Y .+ W ) C_ A )
16 3 4 6 15 syl3anc 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> ( Y .+ W ) C_ A )
17 1 2 paddass 
 |-  ( ( K e. HL /\ ( X C_ A /\ Z C_ A /\ ( Y .+ W ) C_ A ) ) -> ( ( X .+ Z ) .+ ( Y .+ W ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
18 3 10 5 16 17 syl13anc 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> ( ( X .+ Z ) .+ ( Y .+ W ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
19 9 14 18 3eqtr4d 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A ) /\ ( Z C_ A /\ W C_ A ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )