Theorem kbfval 31460

Description: The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐵 ∣ in Dirac notation. See df-kb 31359. (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 7419	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ·ih 𝑧) ·ℎ 𝑦) = ((𝑥 ·ih 𝑧) ·ℎ 𝐴))
2	1	mpteq2dv 5250	. 2 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝑦)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝐴)))
3		oveq2 7419	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑥 ·ih 𝑧) = (𝑥 ·ih 𝐵))
4	3	oveq1d 7426	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑥 ·ih 𝑧) ·ℎ 𝐴) = ((𝑥 ·ih 𝐵) ·ℎ 𝐴))
5	4	mpteq2dv 5250	. 2 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))
6		df-kb 31359	. 2 ⊢ ketbra = (𝑦 ∈ ℋ, 𝑧 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝑧) ·ℎ 𝑦)))
7		ax-hilex 30507	. . 3 ⊢ ℋ ∈ V
8	7	mptex 7227	. 2 ⊢ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)) ∈ V
9	2, 5, 6, 8	ovmpo 7570	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5231  (class class class)co 7411   ℋchba 30427   ·_ℎ csm 30429   ·_ih csp 30430   ketbra ck 30465

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-hilex 30507

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-kb 31359

This theorem is referenced by:  kbop  31461  kbval  31462  kbmul  31463