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Theorem kbop 29368
 Description: The outer product of two vectors, expressed as ∣ 𝐴⟩ ⟨𝐵 ∣ in Dirac notation, is an operator. (Contributed by NM, 30-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
kbop ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)

Proof of Theorem kbop
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hicl 28493 . . . . 5 ((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑥 ·ih 𝐵) ∈ ℂ)
2 hvmulcl 28426 . . . . 5 (((𝑥 ·ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) · 𝐴) ∈ ℋ)
31, 2sylan 577 . . . 4 (((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) · 𝐴) ∈ ℋ)
43an31s 646 . . 3 (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐵) · 𝐴) ∈ ℋ)
54fmpttd 6635 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)): ℋ⟶ ℋ)
6 kbfval 29367 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
76feq1d 6264 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ketbra 𝐵): ℋ⟶ ℋ ↔ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)): ℋ⟶ ℋ))
85, 7mpbird 249 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2166   ↦ cmpt 4953  ⟶wf 6120  (class class class)co 6906  ℂcc 10251   ℋchba 28332   ·ℎ csm 28334   ·ih csp 28335   ketbra ck 28370 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pr 5128  ax-hilex 28412  ax-hfvmul 28418  ax-hfi 28492 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-kb 29266 This theorem is referenced by:  kbpj  29371  kbass2  29532  kbass5  29535
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