Theorem braval 32552

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 32551	. . 3 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
2	1	fveq1d 6888	. 2 ⊢ (𝐴 ∈ ℋ → ((bra‘𝐴)‘𝐵) = ((𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))‘𝐵))
3		oveq1 7428	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ·ih 𝐴) = (𝐵 ·ih 𝐴))
4		eqid 2760	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))
5		ovex 7454	. . 3 ⊢ (𝐵 ·ih 𝐴) ∈ V
6	3, 4, 5	fvmpt 6994	. 2 ⊢ (𝐵 ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))‘𝐵) = (𝐵 ·ih 𝐴))
7	2, 6	sylan9eq 2815	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ‘cfv 6539  (class class class)co 7421   ℋchba 31527   ·_ih csp 31530  bracbr 31564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-hilex 31607

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-bra 32458

This theorem is used by:  braadd  32553  bramul  32554  brafnmul  32559  branmfn  32713  rnbra  32715  bra11  32716  cnvbraval  32718  kbass1  32724  kbass2  32725  kbass6  32729