Theorem braval 29721

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 29720	. . 3 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
2	1	fveq1d 6672	. 2 ⊢ (𝐴 ∈ ℋ → ((bra‘𝐴)‘𝐵) = ((𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))‘𝐵))
3		oveq1 7163	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ·ih 𝐴) = (𝐵 ·ih 𝐴))
4		eqid 2821	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))
5		ovex 7189	. . 3 ⊢ (𝐵 ·ih 𝐴) ∈ V
6	3, 4, 5	fvmpt 6768	. 2 ⊢ (𝐵 ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))‘𝐵) = (𝐵 ·ih 𝐴))
7	2, 6	sylan9eq 2876	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156   ℋchba 28696   ·_ih csp 28699  bracbr 28733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-hilex 28776

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-bra 29627

This theorem is referenced by:  braadd  29722  bramul  29723  brafnmul  29728  branmfn  29882  rnbra  29884  bra11  29885  cnvbraval  29887  kbass1  29893  kbass2  29894  kbass6  29898