Theorem brafval 30206

Description: The bra of a vector, expressed as ⟨𝐴 ∣ in Dirac notation. See df-bra 30113. (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 7263	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥 ·ih 𝑦) = (𝑥 ·ih 𝐴))
2	1	mpteq2dv 5172	. 2 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
3		df-bra 30113	. 2 ⊢ bra = (𝑦 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)))
4		ax-hilex 29262	. . 3 ⊢ ℋ ∈ V
5	4	mptex 7081	. 2 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) ∈ V
6	2, 3, 5	fvmpt 6857	1 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255   ℋchba 29182   ·_ih csp 29185  bracbr 29219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-bra 30113

This theorem is referenced by:  braval  30207  brafn  30210  bra0  30213  brafnmul  30214