Theorem crosspv3d

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

	Ref	Expression
Hypotheses	crosspd.1	$$⊢\phi \to A\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
	crosspd.2	$$⊢\phi \to B\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
Assertion	crosspv3d	Could not format assertion : No typesetting found for |- ( ph -> ( ( A crossp B ) ` 3 ) = ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		crosspd.1	$$⊢\phi \to A\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
2		crosspd.2	$$⊢\phi \to B\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
3		id	$$⊢k=1\to k=1$$
4		1ne3	$$⊢1\ne 3$$
5	4	a1i	$$⊢k=1\to 1\ne 3$$
6	3 5	eqnetrd	$$⊢k=1\to k\ne 3$$
7	6	necon2bi	$$⊢k=3\to \neg k=1$$
8	7	iffalsed	$$⊢k=3\to if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)=if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)$$
9		id	$$⊢k=2\to k=2$$
10		2ne3	$$⊢2\ne 3$$
11	10	a1i	$$⊢k=2\to 2\ne 3$$
12	9 11	eqnetrd	$$⊢k=2\to k\ne 3$$
13	12	necon2bi	$$⊢k=3\to \neg k=2$$
14	13	iffalsed	$$⊢k=3\to if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)=A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)$$
15	8 14	eqtrd	$$⊢k=3\to if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)=A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)$$
16		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 |-
17	1 2 16	syl2anc	Could not format ( 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 ) ) ) ) ) ) ) : No typesetting found for |- ( 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 ) ) ) ) ) ) ) with typecode |-
18		3elfz13	$$⊢3\in \left(1\dots 3\right)$$
19	18	a1i	$$⊢\phi \to 3\in \left(1\dots 3\right)$$
20	1 2	crosspcle3d	$$⊢\phi \to A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\in ℝ$$
21	15 17 19 20	fvmptd4	Could not format ( ph -> ( ( A crossp B ) ` 3 ) = ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) : No typesetting found for |- ( ph -> ( ( A crossp B ) ` 3 ) = ( ( ( A ` 1 ) x. ( B ` 2 ) ) - ( ( A ` 2 ) x. ( B ` 1 ) ) ) ) with typecode |-