his7 - Hilbert Space Explorer

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Theorem his7 27947

Description: Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
his7	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 +_ℎ 𝐶)) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐶)))

**Proof of Theorem his7**
Step	Hyp	Ref	Expression
1		ax-his2 27940	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴)))
2	1	fveq2d 6195	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 +_ℎ 𝐶) ·_ih 𝐴)) = (∗‘((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴))))
3		hicl 27937	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih 𝐴) ∈ ℂ)
4		hicl 27937	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 ·_ih 𝐴) ∈ ℂ)
5		cjadd 13881	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐴) ∈ ℂ ∧ (𝐶 ·_ih 𝐴) ∈ ℂ) → (∗‘((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) + (∗‘(𝐶 ·_ih 𝐴))))
6	3, 4, 5	syl2an 494	. . . . 5 ⊢ (((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ)) → (∗‘((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) + (∗‘(𝐶 ·_ih 𝐴))))
7	6	3impdir 1382	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) + (∗‘(𝐶 ·_ih 𝐴))))
8	2, 7	eqtrd 2656	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 +_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) + (∗‘(𝐶 ·_ih 𝐴))))
9	8	3comr 1273	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐵 +_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) + (∗‘(𝐶 ·_ih 𝐴))))
10		hvaddcl 27869	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
11		ax-his1 27939	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +_ℎ 𝐶) ∈ ℋ) → (𝐴 ·_ih (𝐵 +_ℎ 𝐶)) = (∗‘((𝐵 +_ℎ 𝐶) ·_ih 𝐴)))
12	10, 11	sylan2 491	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝐴 ·_ih (𝐵 +_ℎ 𝐶)) = (∗‘((𝐵 +_ℎ 𝐶) ·_ih 𝐴)))
13	12	3impb 1260	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 +_ℎ 𝐶)) = (∗‘((𝐵 +_ℎ 𝐶) ·_ih 𝐴)))
14		ax-his1 27939	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
15	14	3adant3 1081	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
16		ax-his1 27939	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
17	16	3adant2 1080	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
18	15, 17	oveq12d 6668	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐶)) = ((∗‘(𝐵 ·_ih 𝐴)) + (∗‘(𝐶 ·_ih 𝐴))))
19	9, 13, 18	3eqtr4d 2666	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 +_ℎ 𝐶)) = ((𝐴 ·_ih 𝐵) + (𝐴 ·_ih 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 + caddc 9939 ∗ccj 13836 ℋchil 27776 +_ℎ cva 27777 ·_ih csp 27779

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-pre-mulgt0 10013 ax-hfvadd 27857 ax-hfi 27936 ax-his1 27939 ax-his2 27940

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-rmo 2920 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-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-cj 13839 df-re 13840 df-im 13841

This theorem is referenced by: normlem0 27966 normlem8 27974 pjadjii 28533 lnopunilem1 28869 hmops 28879 cnlnadjlem6 28931 adjlnop 28945 adjadd 28952 hstoh 29091

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