Theorem brafval 32102

Description: The bra of a vector, expressed as ⟨𝐴 ∣ in Dirac notation. See df-bra 32009. (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 7398	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥 ·ih 𝑦) = (𝑥 ·ih 𝐴))
2	1	mpteq2dv 5191	. 2 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
3		df-bra 32009	. 2 ⊢ bra = (𝑦 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)))
4		ax-hilex 31158	. . 3 ⊢ ℋ ∈ V
5	4	mptex 7201	. 2 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) ∈ V
6	2, 3, 5	fvmpt 6969	1 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141   ↦ cmpt 5178  ‘cfv 6515  (class class class)co 7390   ℋchba 31078   ·_ih csp 31081  bracbr 31115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-hilex 31158

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-bra 32009

This theorem is referenced by:  braval  32103  brafn  32106  bra0  32109  brafnmul  32110