hiassdi - Hilbert Space Explorer

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Theorem hiassdi 31604

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 31526	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		ax-his2 31596	. . . 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 1138	. . 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 592	. 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 31597	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
6	5	3expa 1136	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
7	6	adantrl 729	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐷) = (𝐴 · (𝐵 ·_ih 𝐷)))
8	7	oveq1d 7431	. 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 2795	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 (class class class)co 7416 ℂcc 11147 + caddc 11152 · cmul 11154 ℋchba 31432 +_ℎ cva 31433 ·_ℎ csm 31434 ·_ih csp 31435

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-hfvmul 31518 ax-his2 31596 ax-his3 31597

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-fv 6543 df-ov 7419

This theorem is used by: unoplin 32433 hmoplin 32455

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