Theorem brafval 31929

Description: The bra of a vector, expressed as ⟨𝐴 ∣ in Dirac notation. See df-bra 31836. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
brafval	⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem brafval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7418	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥 ·ih 𝑦) = (𝑥 ·ih 𝐴))
2	1	mpteq2dv 5220	. 2 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
3		df-bra 31836	. 2 ⊢ bra = (𝑦 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)))
4		ax-hilex 30985	. . 3 ⊢ ℋ ∈ V
5	4	mptex 7220	. 2 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) ∈ V
6	2, 3, 5	fvmpt 6991	1 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5206  ‘cfv 6536  (class class class)co 7410   ℋchba 30905   ·_ih csp 30908  bracbr 30942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-hilex 30985

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-bra 31836

This theorem is referenced by:  braval  31930  brafn  31933  bra0  31936  brafnmul  31937