Theorem kbass4

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	kbass4	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) · ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ) = ( ( bra ‘ 𝐴 ) ‘ ( ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ·ℎ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bracl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ∈ ℂ )
2		bracl	⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) → ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ∈ ℂ )
3		mulcom	⊢ ( ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ∈ ℂ ∧ ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ∈ ℂ ) → ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) · ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ) = ( ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) · ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ) )
4	1 2 3	syl2an	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) · ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ) = ( ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) · ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ) )
5		bralnfn	⊢ ( 𝐴 ∈ ℋ → ( bra ‘ 𝐴 ) ∈ LinFn )
6	5	ad2antrr	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( bra ‘ 𝐴 ) ∈ LinFn )
7	2	adantl	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ∈ ℂ )
8		simplr	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → 𝐵 ∈ ℋ )
9		lnfnmul	⊢ ( ( ( bra ‘ 𝐴 ) ∈ LinFn ∧ ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ·ℎ 𝐵 ) ) = ( ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) · ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ) )
10	6 7 8 9	syl3anc	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( bra ‘ 𝐴 ) ‘ ( ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ·ℎ 𝐵 ) ) = ( ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) · ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) ) )
11	4 10	eqtr4d	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) · ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ) = ( ( bra ‘ 𝐴 ) ‘ ( ( ( bra ‘ 𝐶 ) ‘ 𝐷 ) ·ℎ 𝐵 ) ) )