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Theorem kbop 28661
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 27786 . . . . 5 ((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑥 ·ih 𝐵) ∈ ℂ)
2 hvmulcl 27719 . . . . 5 (((𝑥 ·ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) · 𝐴) ∈ ℋ)
31, 2sylan 488 . . . 4 (((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) · 𝐴) ∈ ℋ)
43an31s 847 . . 3 (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐵) · 𝐴) ∈ ℋ)
5 eqid 2621 . . 3 (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴))
64, 5fmptd 6340 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)): ℋ⟶ ℋ)
7 kbfval 28660 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
87feq1d 5987 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ketbra 𝐵): ℋ⟶ ℋ ↔ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)): ℋ⟶ ℋ))
96, 8mpbird 247 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  cmpt 4673  wf 5843  (class class class)co 6604  cc 9878  chil 27625   · csm 27627   ·ih csp 27628   ketbra ck 27663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-hilex 27705  ax-hfvmul 27711  ax-hfi 27785
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-kb 28559
This theorem is referenced by:  kbpj  28664  kbass2  28825  kbass5  28828
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