hiassdi - Hilbert Space Explorer

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Theorem hiassdi 31693

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 31615	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		ax-his2 31685	. . . 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 31686	. . . . 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 7435	. 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 1569 ∈ wcel 2144 (class class class)co 7420 ℂcc 11198 + caddc 11203 · cmul 11205 ℋchba 31521 +_ℎ cva 31522 ·_ℎ csm 31523 ·_ih csp 31524

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-hfvmul 31607 ax-his2 31685 ax-his3 31686

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fv 6546 df-ov 7423

This theorem is used by: unoplin 32522 hmoplin 32544

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