Theorem kbfval 29731

Description: The outer product of two vectors, expressed as ∣ 𝐴⟩ ⟨𝐵 ∣ in Dirac notation. See df-kb 29630. (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) ·ℎ 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.

Step	Hyp	Ref	Expression
1		oveq2 7166	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ·ih 𝑧) ·ℎ 𝑦) = ((𝑥 ·ih 𝑧) ·ℎ 𝐴))
2	1	mpteq2dv 5164	. 2 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝐴)))
3		oveq2 7166	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑥 ·ih 𝑧) = (𝑥 ·ih 𝐵))
4	3	oveq1d 7173	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑥 ·ih 𝑧) ·ℎ 𝐴) = ((𝑥 ·ih 𝐵) ·ℎ 𝐴))
5	4	mpteq2dv 5164	. 2 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))
6		df-kb 29630	. 2 ⊢ ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝑦)))
7		ax-hilex 28778	. . 3 ⊢ ℋ ∈ V
8	7	mptex 6988	. 2 ⊢ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)) ∈ V
9	2, 5, 6, 8	ovmpo 7312	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ↦ cmpt 5148  (class class class)co 7158   ℋchba 28698   ·_ℎ csm 28700   ·_ih csp 28701   ketbra ck 28736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-hilex 28778

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-kb 29630

This theorem is referenced by:  kbop  29732  kbval  29733  kbmul  29734