Theorem crosspv1i

Description: Value of the first component of the cross product. (Contributed by Jiamin Zhao, 31-Jul-2026)

	Ref	Expression
Hypotheses	crossp.1	$$⊢A\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
	crossp.2	$$⊢B\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
Assertion	crosspv1i	Could not format assertion : No typesetting found for |- ( ( A crossp B ) ` 1 ) = ( ( ( A ` 2 ) x. ( B ` 3 ) ) - ( ( A ` 3 ) x. ( B ` 2 ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		crossp.1	$$⊢A\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
2		crossp.2	$$⊢B\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
3		crosspval	Could not format ( ( 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 ) ) ) ) ) ) ) : No typesetting found for |- ( ( 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 ) ) ) ) ) ) ) with typecode |-
4	1 2 3	mp2an	Could not format ( 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 ) ) ) ) ) ) : No typesetting found for |- ( 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 ) ) ) ) ) ) with typecode |-
5	4	fveq1i	Could not format ( ( A crossp B ) ` 1 ) = ( ( 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 ) ) ) ) ) ) ` 1 ) : No typesetting found for |- ( ( A crossp B ) ` 1 ) = ( ( 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 ) ) ) ) ) ) ` 1 ) with typecode |-
6		1elfz13	$$⊢1\in \left(1\dots 3\right)$$
7	1 2	crosspcle1i	$$⊢A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right)\in ℝ$$
8		iftrue	$$⊢k=1\to if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)=A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right)$$
9		eqid	$$⊢\left(k\in \left(1\dots 3\right)⟼if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\right)=\left(k\in \left(1\dots 3\right)⟼if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\right)$$
10	8 9	fvmptg	$$⊢\left(1\in \left(1\dots 3\right)\wedge A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right)\in ℝ\right)\to \left(k\in \left(1\dots 3\right)⟼if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\right)\left(1\right)=A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right)$$
11	6 7 10	mp2an	$$⊢\left(k\in \left(1\dots 3\right)⟼if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\right)\left(1\right)=A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right)$$
12	5 11	eqtri	Could not format ( ( A crossp B ) ` 1 ) = ( ( ( A ` 2 ) x. ( B ` 3 ) ) - ( ( A ` 3 ) x. ( B ` 2 ) ) ) : No typesetting found for |- ( ( A crossp B ) ` 1 ) = ( ( ( A ` 2 ) x. ( B ` 3 ) ) - ( ( A ` 3 ) x. ( B ` 2 ) ) ) with typecode |-