Theorem kbass3

Description: Dirac bra-ket associative law <. A | B >. <. C | D >. = ( <. A | B >. <. C | ) | D >. . (Contributed by NM, 30-May-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	kbass3	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) · ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ) = ( ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ·fn ( bra ‘ 𝐶 ) ) ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		bracl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ∈ ℂ )
2	1	adantr	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ∈ ℂ )
3		brafn	⊢ ( 𝐶 ∈ ℋ → ( bra ‘ 𝐶 ) : ℋ ⟶ ℂ )
4	3	ad2antrl	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( bra ‘ 𝐶 ) : ℋ ⟶ ℂ )
5		simprr	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → 𝐷 ∈ ℋ )
6		hfmval	⊢ ( ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ∈ ℂ ∧ ( bra ‘ 𝐶 ) : ℋ ⟶ ℂ ∧ 𝐷 ∈ ℋ ) → ( ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ·fn ( bra ‘ 𝐶 ) ) ‘ 𝐷 ) = ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) · ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ) )
7	2 4 5 6	syl3anc	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ·fn ( bra ‘ 𝐶 ) ) ‘ 𝐷 ) = ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) · ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ) )
8	7	eqcomd	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) · ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ) = ( ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ·fn ( bra ‘ 𝐶 ) ) ‘ 𝐷 ) )