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Theorem kbfval 29739
Description: The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation. See df-kb 29638. (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 7147 . . 3 (𝑦 = 𝐴 → ((𝑥 ·ih 𝑧) · 𝑦) = ((𝑥 ·ih 𝑧) · 𝐴))
21mpteq2dv 5129 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝐴)))
3 oveq2 7147 . . . 4 (𝑧 = 𝐵 → (𝑥 ·ih 𝑧) = (𝑥 ·ih 𝐵))
43oveq1d 7154 . . 3 (𝑧 = 𝐵 → ((𝑥 ·ih 𝑧) · 𝐴) = ((𝑥 ·ih 𝐵) · 𝐴))
54mpteq2dv 5129 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
6 df-kb 29638 . 2 ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) · 𝑦)))
7 ax-hilex 28786 . . 3 ℋ ∈ V
87mptex 6967 . 2 (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)) ∈ V
92, 5, 6, 8ovmpo 7293 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 399   = wceq 1538  wcel 2112  cmpt 5113  (class class class)co 7139  chba 28706   · csm 28708   ·ih csp 28709   ketbra ck 28744
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-hilex 28786
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-kb 29638
This theorem is referenced by:  kbop  29740  kbval  29741  kbmul  29742
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