Theorem kbval 32048

Description: The value of the operator resulting from the outer product ∣ 𝐴⟩ ⟨𝐵 ∣ of two vectors. Equation 8.1 of [Prugovecki] p. 376. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
kbval	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))

Proof of Theorem kbval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kbfval 32046	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))
2	1	fveq1d 6846	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))‘𝐶))
3		oveq1 7377	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ·ih 𝐵) = (𝐶 ·ih 𝐵))
4	3	oveq1d 7385	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
5		eqid 2737	. . . 4 ⊢ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))
6		ovex 7403	. . . 4 ⊢ ((𝐶 ·ih 𝐵) ·ℎ 𝐴) ∈ V
7	4, 5, 6	fvmpt 6951	. . 3 ⊢ (𝐶 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
8	2, 7	sylan9eq 2792	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
9	8	3impa 1110	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5181  ‘cfv 6502  (class class class)co 7370   ℋchba 31013   ·_ℎ csm 31015   ·_ih csp 31016   ketbra ck 31051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-hilex 31093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-kb 31945

This theorem is referenced by:  kbpj  32050  kbass1  32210  kbass2  32211  kbass5  32214  kbass6  32215