Theorem kbass1 32145

Description: Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ ) ∣ 𝐶⟩ = ∣ 𝐴⟩(⟨𝐵 ∣ 𝐶⟩), i.e., the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since ⟨𝐵 ∣ 𝐶⟩ is a complex number, it is the first argument in the inner product ·_ℎ that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
kbass1	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) ·ℎ 𝐴))

Proof of Theorem kbass1

Step	Hyp	Ref	Expression
1		kbval 31983	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
2		braval 31973	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐵)‘𝐶) = (𝐶 ·ih 𝐵))
3	2	3adant1 1129	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐵)‘𝐶) = (𝐶 ·ih 𝐵))
4	3	oveq1d 7446	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐵)‘𝐶) ·ℎ 𝐴) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
5	1, 4	eqtr4d 2778	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) ·ℎ 𝐴))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431   ℋchba 30948   ·_ℎ csm 30950   ·_ih csp 30951  bracbr 30985   ketbra ck 30986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-hilex 31028

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-bra 31879  df-kb 31880

This theorem is referenced by: (None)