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Theorem kbfval 29116
Description: The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation. See df-kb 29015. (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 6817 . . 3 (𝑦 = 𝐴 → ((𝑥 ·ih 𝑧) · 𝑦) = ((𝑥 ·ih 𝑧) · 𝐴))
21mpteq2dv 4893 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝐴)))
3 oveq2 6817 . . . 4 (𝑧 = 𝐵 → (𝑥 ·ih 𝑧) = (𝑥 ·ih 𝐵))
43oveq1d 6824 . . 3 (𝑧 = 𝐵 → ((𝑥 ·ih 𝑧) · 𝐴) = ((𝑥 ·ih 𝐵) · 𝐴))
54mpteq2dv 4893 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
6 df-kb 29015 . 2 ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝑦)))
7 ax-hilex 28161 . . 3 ℋ ∈ V
87mptex 6646 . 2 (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)) ∈ V
92, 5, 6, 8ovmpt2 6957 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 383   = wceq 1628  wcel 2135  cmpt 4877  (class class class)co 6809  chil 28081   · csm 28083   ·ih csp 28084   ketbra ck 28119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pr 5051  ax-hilex 28161
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-kb 29015
This theorem is referenced by:  kbop  29117  kbval  29118  kbmul  29119
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