Metamath Proof Explorer

Theorem crosspv2d

Description: Value of the second component of the cross product. (Contributed by Jiamin Zhao, 12-Aug-2026)

Ref Expression
Hypotheses crosspd.1 
|- ( ph -> A e. ( RR ^m ( 1 ... 3 ) ) )
crosspd.2 
|- ( ph -> B e. ( RR ^m ( 1 ... 3 ) ) )
Assertion crosspv2d 
|- ( ph -> ( ( A crossp B ) ` 2 ) = ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) )

Proof

Step Hyp Ref Expression
1 crosspd.1 
 |-  ( ph -> A e. ( RR ^m ( 1 ... 3 ) ) )
2 crosspd.2 
 |-  ( ph -> B e. ( RR ^m ( 1 ... 3 ) ) )
3 id 
 |-  ( k = 1 -> k = 1 )
4 1ne2 
 |-  1 =/= 2
5 4 a1i 
 |-  ( k = 1 -> 1 =/= 2 )
6 3 5 eqnetrd 
 |-  ( k = 1 -> k =/= 2 )
7 6 necon2bi 
 |-  ( k = 2 -> -. k = 1 )
8 7 iffalsed 
 |-  ( k = 2 -> if ( k = 1 , ( ( ( A ` 2 ) x. ( B ` 3 ) ) - ( ( A ` 3 ) x. ( B ` 2 ) ) ) , if ( k = 2 , ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) , ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) ) = if ( k = 2 , ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) , ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) )
9 iftrue 
 |-  ( k = 2 -> if ( k = 2 , ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) , ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) = ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) )
10 8 9 eqtrd 
 |-  ( k = 2 -> if ( k = 1 , ( ( ( A ` 2 ) x. ( B ` 3 ) ) - ( ( A ` 3 ) x. ( B ` 2 ) ) ) , if ( k = 2 , ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) , ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) ) = ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) )
11 crosspval 
 |-  ( ( A e. ( RR ^m ( 1 ... 3 ) ) /\ B e. ( RR ^m ( 1 ... 3 ) ) ) -> ( A crossp B ) = ( k e. ( 1 ... 3 ) |-> if ( k = 1 , ( ( ( A ` 2 ) x. ( B ` 3 ) ) - ( ( A ` 3 ) x. ( B ` 2 ) ) ) , if ( k = 2 , ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) , ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) ) ) )
12 1 2 11 syl2anc 
 |-  ( ph -> ( A crossp B ) = ( k e. ( 1 ... 3 ) |-> if ( k = 1 , ( ( ( A ` 2 ) x. ( B ` 3 ) ) - ( ( A ` 3 ) x. ( B ` 2 ) ) ) , if ( k = 2 , ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) , ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) ) ) )
13 2elfz13 
 |-  2 e. ( 1 ... 3 )
14 13 a1i 
 |-  ( ph -> 2 e. ( 1 ... 3 ) )
15 1 2 crosspcle2d 
 |-  ( ph -> ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) e. RR )
16 10 12 14 15 fvmptd4 
 |-  ( ph -> ( ( A crossp B ) ` 2 ) = ( ( ( A ` 3 ) x. ( B ` 1 ) ) - ( ( A ` 1 ) x. ( B ` 3 ) ) ) )