hiassdi - Hilbert Space Explorer

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Theorem hiassdi 28871

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 28793	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		ax-his2 28863	. . . 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 1116	. . 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 582	. 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 28864	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
6	5	3expa 1114	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
7	6	adantrl 714	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
8	7	oveq1d 7174	. 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 2859	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 398 = wceq 1536 ∈ wcel 2113 (class class class)co 7159 ℂcc 10538 + caddc 10543 · cmul 10545 ℋchba 28699 +_ℎ cva 28700 ·_ℎ csm 28701 ·_ih csp 28702

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-hfvmul 28785 ax-his2 28863 ax-his3 28864

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162

This theorem is referenced by: unoplin 29700 hmoplin 29722

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