his2sub - Hilbert Space Explorer

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Theorem his2sub 27949

Description: Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
his2sub	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))

**Proof of Theorem his2sub**
Step	Hyp	Ref	Expression
1		hvsubval 27873	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
2	1	oveq1d 6665	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
3	2	3adant3 1081	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
4		neg1cn 11124	. . . . 5 ⊢ -1 ∈ ℂ
5		hvmulcl 27870	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
6	4, 5	mpan 706	. . . 4 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
7		ax-his2 27940	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (-1 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)))
8	6, 7	syl3an2 1360	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)))
9		ax-his3 27941	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
10	4, 9	mp3an1 1411	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
11		hicl 27937	. . . . . . 7 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
12	11	mulm1d 10482	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶))
13	10, 12	eqtrd 2656	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶))
14	13	oveq2d 6666	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
15	14	3adant1 1079	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
16	8, 15	eqtrd 2656	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
17		hicl 27937	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
18	17	3adant2 1080	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
19	11	3adant1 1079	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
20	18, 19	negsubd 10398	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))
21	3, 16, 20	3eqtrd 2660	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 -cneg 10267 ℋchil 27776 +_ℎ cva 27777 ·_ℎ csm 27778 ·_ih csp 27779 −_ℎ cmv 27782

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-hfvmul 27862 ax-hfi 27936 ax-his2 27940 ax-his3 27941

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-hvsub 27828

This theorem is referenced by: his2sub2 27950 hi2eq 27962 pjhthlem1 28250 h1de2i 28412 pjdifnormii 28542 lnopeqi 28867 riesz3i 28921 leop2 28983 hmopidmpji 29011 pjssposi 29031 pjclem4 29058 pj3si 29066

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