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Theorem kbfval 29162
Description: The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation. See df-kb 29061. (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 6892 . . 3 (𝑦 = 𝐴 → ((𝑥 ·ih 𝑧) · 𝑦) = ((𝑥 ·ih 𝑧) · 𝐴))
21mpteq2dv 4950 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝐴)))
3 oveq2 6892 . . . 4 (𝑧 = 𝐵 → (𝑥 ·ih 𝑧) = (𝑥 ·ih 𝐵))
43oveq1d 6899 . . 3 (𝑧 = 𝐵 → ((𝑥 ·ih 𝑧) · 𝐴) = ((𝑥 ·ih 𝐵) · 𝐴))
54mpteq2dv 4950 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
6 df-kb 29061 . 2 ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝑦)))
7 ax-hilex 28207 . . 3 ℋ ∈ V
87mptex 6721 . 2 (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)) ∈ V
92, 5, 6, 8ovmpt2 7036 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 1637  wcel 2157  cmpt 4934  (class class class)co 6884  chil 28127   · csm 28129   ·ih csp 28130   ketbra ck 28165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pr 5109  ax-hilex 28207
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-kb 29061
This theorem is referenced by:  kbop  29163  kbval  29164  kbmul  29165
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