Theorem kbop 32039

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 32038	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))
2		hicl 31166	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑥 ·ih 𝐵) ∈ ℂ)
3		hvmulcl 31099	. . . 4 ⊢ (((𝑥 ·ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
4	2, 3	sylan 581	. . 3 ⊢ (((𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
5	4	an31s 655	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) ∈ ℋ)
6	1, 5	fmpt3d 7062	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ⟶wf 6488  (class class class)co 7360  ℂcc 11027   ℋchba 31005   ·_ℎ csm 31007   ·_ih csp 31008   ketbra ck 31043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-hilex 31085  ax-hfvmul 31091  ax-hfi 31165

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-kb 31937

This theorem is referenced by:  kbpj  32042  kbass2  32203  kbass5  32206