his2sub2 - Hilbert Space Explorer

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Theorem his2sub2 31179

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 31178	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴)))
2	1	fveq2d 6838	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))))
3		hicl 31166	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih 𝐴) ∈ ℂ)
4		hicl 31166	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 ·_ih 𝐴) ∈ ℂ)
5		cjsub 15102	. . . . . 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 1353	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
8	2, 7	eqtrd 2772	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
9	8	3comr 1126	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
10		hvsubcl 31103	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		ax-his1 31168	. . . 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 1115	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
14		ax-his1 31168	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
15	14	3adant3 1133	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
16		ax-his1 31168	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
17	16	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
18	15, 17	oveq12d 7378	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐵) − (𝐴 ·_ih 𝐶)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
19	9, 13, 18	3eqtr4d 2782	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 − cmin 11368 ∗ccj 15049 ℋchba 31005 ·_ih csp 31008 −_ℎ cmv 31011

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-hfvadd 31086 ax-hfvmul 31091 ax-hfi 31165 ax-his1 31168 ax-his2 31169 ax-his3 31170

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-cj 15052 df-re 15053 df-im 15054 df-hvsub 31057

This theorem is referenced by: pjhthlem1 31477 riesz4i 32149

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