hiassdi - Hilbert Space Explorer

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Theorem hiassdi 31082

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 31004	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		ax-his2 31074	. . . 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 1120	. . 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 31075	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
6	5	3expa 1118	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
7	6	adantrl 716	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
8	7	oveq1d 7370	. 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 2768	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 395 = wceq 1541 ∈ wcel 2113 (class class class)co 7355 ℂcc 11014 + caddc 11019 · cmul 11021 ℋchba 30910 +_ℎ cva 30911 ·_ℎ csm 30912 ·_ih csp 30913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pr 5374 ax-hfvmul 30996 ax-his2 31074 ax-his3 31075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-ss 3916 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-fv 6497 df-ov 7358

This theorem is referenced by: unoplin 31911 hmoplin 31933

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