his2sub2 - Hilbert Space Explorer

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Theorem his2sub2 28278

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 28277	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴)))
2	1	fveq2d 6412	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))))
3		hicl 28265	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih 𝐴) ∈ ℂ)
4		hicl 28265	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 ·_ih 𝐴) ∈ ℂ)
5		cjsub 14112	. . . . . 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 585	. . . . 5 ⊢ (((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ)) → (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
7	6	3impdir 1453	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
8	2, 7	eqtrd 2840	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
9	8	3comr 1148	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
10		hvsubcl 28202	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		ax-his1 28267	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 −_ℎ 𝐶) ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
12	10, 11	sylan2 582	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
13	12	3impb 1136	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
14		ax-his1 28267	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
15	14	3adant3 1155	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
16		ax-his1 28267	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
17	16	3adant2 1154	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
18	15, 17	oveq12d 6892	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐵) − (𝐴 ·_ih 𝐶)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
19	9, 13, 18	3eqtr4d 2850	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 1100 = wceq 1637 ∈ wcel 2156 ‘cfv 6101 (class class class)co 6874 ℂcc 10219 − cmin 10551 ∗ccj 14059 ℋchil 28104 ·_ih csp 28107 −_ℎ cmv 28110

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2068 ax-7 2104 ax-8 2158 ax-9 2165 ax-10 2185 ax-11 2201 ax-12 2214 ax-13 2420 ax-ext 2784 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5096 ax-un 7179 ax-resscn 10278 ax-1cn 10279 ax-icn 10280 ax-addcl 10281 ax-addrcl 10282 ax-mulcl 10283 ax-mulrcl 10284 ax-mulcom 10285 ax-addass 10286 ax-mulass 10287 ax-distr 10288 ax-i2m1 10289 ax-1ne0 10290 ax-1rid 10291 ax-rnegex 10292 ax-rrecex 10293 ax-cnre 10294 ax-pre-lttri 10295 ax-pre-lttrn 10296 ax-pre-ltadd 10297 ax-pre-mulgt0 10298 ax-hfvadd 28185 ax-hfvmul 28190 ax-hfi 28264 ax-his1 28267 ax-his2 28268 ax-his3 28269

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2061 df-eu 2634 df-mo 2635 df-clab 2793 df-cleq 2799 df-clel 2802 df-nfc 2937 df-ne 2979 df-nel 3082 df-ral 3101 df-rex 3102 df-reu 3103 df-rmo 3104 df-rab 3105 df-v 3393 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4117 df-if 4280 df-pw 4353 df-sn 4371 df-pr 4373 df-op 4377 df-uni 4631 df-iun 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5219 df-po 5232 df-so 5233 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6835 df-ov 6877 df-oprab 6878 df-mpt2 6879 df-er 7979 df-en 8193 df-dom 8194 df-sdom 8195 df-pnf 10361 df-mnf 10362 df-xr 10363 df-ltxr 10364 df-le 10365 df-sub 10553 df-neg 10554 df-div 10970 df-2 11364 df-cj 14062 df-re 14063 df-im 14064 df-hvsub 28156

This theorem is referenced by: pjhthlem1 28578 riesz4i 29250

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