Theorem braval 32103

Description: A bra-ket juxtaposition, expressed as ⟨𝐴 ∣ 𝐵⟩ in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
braval	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))

Proof of Theorem braval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brafval 32102	. . 3 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
2	1	fveq1d 6863	. 2 ⊢ (𝐴 ∈ ℋ → ((bra‘𝐴)‘𝐵) = ((𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))‘𝐵))
3		oveq1 7397	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ·ih 𝐴) = (𝐵 ·ih 𝐴))
4		eqid 2761	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))
5		ovex 7423	. . 3 ⊢ (𝐵 ·ih 𝐴) ∈ V
6	3, 4, 5	fvmpt 6969	. 2 ⊢ (𝐵 ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))‘𝐵) = (𝐵 ·ih 𝐴))
7	2, 6	sylan9eq 2816	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = 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:  braadd  32104  bramul  32105  brafnmul  32110  branmfn  32264  rnbra  32266  bra11  32267  cnvbraval  32269  kbass1  32275  kbass2  32276  kbass6  32280