Theorem kbop 32210

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 32209	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))
2		hicl 31337	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑥 ·ih 𝐵) ∈ ℂ)
3		hvmulcl 31270	. . . 4 ⊢ (((𝑥 ·ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
4	2, 3	sylan 591	. . 3 ⊢ (((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
5	4	an31s 666	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
6	1, 5	fmpt3d 7101	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  ⟶wf 6521  (class class class)co 7400  ℂcc 11086   ℋchba 31176   ·_ℎ csm 31178   ·_ih csp 31179   ketbra ck 31214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-hilex 31256  ax-hfvmul 31262  ax-hfi 31336

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-kb 32108

This theorem is referenced by:  kbpj  32213  kbass2  32374  kbass5  32377