Theorem kbop 31639

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.

Step	Hyp	Ref	Expression
1		kbfval 31638	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))
2		hicl 30766	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑥 ·ih 𝐵) ∈ ℂ)
3		hvmulcl 30699	. . . 4 ⊢ (((𝑥 ·ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
4	2, 3	sylan 579	. . 3 ⊢ (((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
5	4	an31s 651	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
6	1, 5	fmpt3d 7117	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105  ⟶wf 6539  (class class class)co 7412  ℂcc 11114   ℋchba 30605   ·_ℎ csm 30607   ·_ih csp 30608   ketbra ck 30643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-hilex 30685  ax-hfvmul 30691  ax-hfi 30765

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-kb 31537

This theorem is referenced by:  kbpj  31642  kbass2  31803  kbass5  31806