Theorem kbval 32096

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 32094	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)))
2	1	fveq1d 6858	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))‘𝐶))
3		oveq1 7392	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ·ih 𝐵) = (𝐶 ·ih 𝐵))
4	3	oveq1d 7400	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ·ih 𝐵) ·ℎ 𝐴) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
5		eqid 2756	. . . 4 ⊢ (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))
6		ovex 7418	. . . 4 ⊢ ((𝐶 ·ih 𝐵) ·ℎ 𝐴) ∈ V
7	4, 5, 6	fvmpt 6964	. . 3 ⊢ (𝐶 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
8	2, 7	sylan9eq 2811	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
9	8	3impa 1118	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ↦ cmpt 5175  ‘cfv 6510  (class class class)co 7385   ℋchba 31061   ·_ℎ csm 31063   ·_ih csp 31064   ketbra ck 31099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-hilex 31141

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-kb 31993

This theorem is referenced by:  kbpj  32098  kbass1  32258  kbass2  32259  kbass5  32262  kbass6  32263