his2sub2 - Hilbert Space Explorer

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Theorem his2sub2 28869

Description: Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
his2sub2	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ih 𝐵) − (𝐴 ·_ih 𝐶)))

**Proof of Theorem his2sub2**
Step	Hyp	Ref	Expression
1		his2sub 28868	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴)))
2	1	fveq2d 6673	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))))
3		hicl 28856	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih 𝐴) ∈ ℂ)
4		hicl 28856	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 ·_ih 𝐴) ∈ ℂ)
5		cjsub 14507	. . . . . 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 597	. . . . 5 ⊢ (((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ)) → (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
7	6	3impdir 1347	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
8	2, 7	eqtrd 2856	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
9	8	3comr 1121	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
10		hvsubcl 28793	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		ax-his1 28858	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 −_ℎ 𝐶) ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
12	10, 11	sylan2 594	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
13	12	3impb 1111	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
14		ax-his1 28858	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
15	14	3adant3 1128	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
16		ax-his1 28858	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
17	16	3adant2 1127	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
18	15, 17	oveq12d 7173	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐵) − (𝐴 ·_ih 𝐶)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
19	9, 13, 18	3eqtr4d 2866	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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 − cmin 10869 ∗ccj 14454 ℋchba 28695 ·_ih csp 28698 −_ℎ cmv 28701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-hfvadd 28776 ax-hfvmul 28781 ax-hfi 28855 ax-his1 28858 ax-his2 28859 ax-his3 28860

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-2 11699 df-cj 14457 df-re 14458 df-im 14459 df-hvsub 28747

This theorem is referenced by: pjhthlem1 29167 riesz4i 29839

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