hiassdi - Hilbert Space Explorer

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Theorem hiassdi 29447

Description: Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hiassdi	⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·_ℎ 𝐵) +_ℎ 𝐶) ·_ih 𝐷) = ((𝐴 · (𝐵 ·_ih 𝐷)) + (𝐶 ·_ih 𝐷)))

**Proof of Theorem hiassdi**
Step	Hyp	Ref	Expression
1		hvmulcl 29369	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		ax-his2 29439	. . . 4 ⊢ (((𝐴 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐴 ·_ℎ 𝐵) +_ℎ 𝐶) ·_ih 𝐷) = (((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) + (𝐶 ·_ih 𝐷)))
3	2	3expb 1119	. . 3 ⊢ (((𝐴 ·_ℎ 𝐵) ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·_ℎ 𝐵) +_ℎ 𝐶) ·_ih 𝐷) = (((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) + (𝐶 ·_ih 𝐷)))
4	1, 3	sylan 580	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·_ℎ 𝐵) +_ℎ 𝐶) ·_ih 𝐷) = (((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) + (𝐶 ·_ih 𝐷)))
5		ax-his3 29440	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
6	5	3expa 1117	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
7	6	adantrl 713	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
8	7	oveq1d 7284	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) + (𝐶 ·_ih 𝐷)) = ((𝐴 · (𝐵 ·_ih 𝐷)) + (𝐶 ·_ih 𝐷)))
9	4, 8	eqtrd 2780	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·_ℎ 𝐵) +_ℎ 𝐶) ·_ih 𝐷) = ((𝐴 · (𝐵 ·_ih 𝐷)) + (𝐶 ·_ih 𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 (class class class)co 7269 ℂcc 10868 + caddc 10873 · cmul 10875 ℋchba 29275 +_ℎ cva 29276 ·_ℎ csm 29277 ·_ih csp 29278

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-hfvmul 29361 ax-his2 29439 ax-his3 29440

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-fv 6439 df-ov 7272

This theorem is referenced by: unoplin 30276 hmoplin 30298

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