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Theorem braval 31826

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 31825	. . 3 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐴)))
2	1	fveq1d 6898	. 2 ⊢ (𝐴 ∈ ℋ → ((bra‘𝐴)‘𝐵) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐴))‘𝐵))
3		oveq1 7426	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ·_ih 𝐴) = (𝐵 ·_ih 𝐴))
4		eqid 2725	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐴))
5		ovex 7452	. . 3 ⊢ (𝐵 ·_ih 𝐴) ∈ V
6	3, 4, 5	fvmpt 7004	. 2 ⊢ (𝐵 ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐴))‘𝐵) = (𝐵 ·_ih 𝐴))
7	2, 6	sylan9eq 2785	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·_ih 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ℋchba 30801 ·_ih csp 30804 bracbr 30838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-hilex 30881

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-bra 31732

This theorem is referenced by: braadd 31827 bramul 31828 brafnmul 31833 branmfn 31987 rnbra 31989 bra11 31990 cnvbraval 31992 kbass1 31998 kbass2 31999 kbass6 32003

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