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Theorem kbfval 28651
Description: The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation. See df-kb 28550. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
kbfval ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem kbfval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6613 . . 3 (𝑦 = 𝐴 → ((𝑥 ·ih 𝑧) · 𝑦) = ((𝑥 ·ih 𝑧) · 𝐴))
21mpteq2dv 4710 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝐴)))
3 oveq2 6613 . . . 4 (𝑧 = 𝐵 → (𝑥 ·ih 𝑧) = (𝑥 ·ih 𝐵))
43oveq1d 6620 . . 3 (𝑧 = 𝐵 → ((𝑥 ·ih 𝑧) · 𝐴) = ((𝑥 ·ih 𝐵) · 𝐴))
54mpteq2dv 4710 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
6 df-kb 28550 . 2 ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝑦)))
7 ax-hilex 27696 . . 3 ℋ ∈ V
87mptex 6441 . 2 (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)) ∈ V
92, 5, 6, 8ovmpt2 6750 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  cmpt 4678  (class class class)co 6605  chil 27616   · csm 27618   ·ih csp 27619   ketbra ck 27654
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-hilex 27696
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-kb 28550
This theorem is referenced by:  kbop  28652  kbval  28653  kbmul  28654
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