Theorem crosspcld

Description: Closure of the cross product: the cross product of two 3-dimensional real coordinate vectors is again such a vector. (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	crosspcld	Could not format assertion : No typesetting found for |- ( ph -> ( A crossp B ) e. ( RR ^m ( 1 ... 3 ) ) ) 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		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	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 |-
5	1 2	crosspclem	$$⊢\phi \to if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\in ℝ$$
6	5	adantr	$$⊢\left(\phi \wedge k\in \left(1\dots 3\right)\right)\to if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\in ℝ$$
7	6	fmpttd	$$⊢\phi \to \left(k\in \left(1\dots 3\right)⟼if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\right):\left(1\dots 3\right)⟶ℝ$$
8		reex	$$⊢ℝ\in \mathrm{V}$$
9		ovex	$$⊢\left(1\dots 3\right)\in \mathrm{V}$$
10	8 9	elmap	$$⊢\left(k\in \left(1\dots 3\right)⟼if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\right)\in \left({ℝ}^{\left(1\dots 3\right)}\right)\leftrightarrow \left(k\in \left(1\dots 3\right)⟼if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\right):\left(1\dots 3\right)⟶ℝ$$
11	7 10	sylibr	$$⊢\phi \to \left(k\in \left(1\dots 3\right)⟼if\left(k=1,A\left(2\right)B\left(3\right)-A\left(3\right)B\left(2\right),if\left(k=2,A\left(3\right)B\left(1\right)-A\left(1\right)B\left(3\right),A\left(1\right)B\left(2\right)-A\left(2\right)B\left(1\right)\right)\right)\right)\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
12	4 11	eqeltrd	Could not format ( ph -> ( A crossp B ) e. ( RR ^m ( 1 ... 3 ) ) ) : No typesetting found for |- ( ph -> ( A crossp B ) e. ( RR ^m ( 1 ... 3 ) ) ) with typecode |-