his2sub - Hilbert Space Explorer

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Theorem his2sub 31181

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 31105	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
2	1	oveq1d 7376	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
3	2	3adant3 1133	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
4		neg1cn 12138	. . . . 5 ⊢ -1 ∈ ℂ
5		hvmulcl 31102	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
6	4, 5	mpan 691	. . . 4 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
7		ax-his2 31172	. . . 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 1165	. . 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 31173	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
10	4, 9	mp3an1 1451	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
11		hicl 31169	. . . . . . 7 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
12	11	mulm1d 11596	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶))
13	10, 12	eqtrd 2772	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶))
14	13	oveq2d 7377	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
15	14	3adant1 1131	. . 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 2772	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
17		hicl 31169	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
18	17	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
19	11	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
20	18, 19	negsubd 11505	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))
21	3, 16, 20	3eqtrd 2776	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 (class class class)co 7361 ℂcc 11030 1c1 11033 + caddc 11035 · cmul 11037 − cmin 11371 -cneg 11372 ℋchba 31008 +_ℎ cva 31009 ·_ℎ csm 31010 ·_ih csp 31011 −_ℎ cmv 31014

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 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-hfvmul 31094 ax-hfi 31168 ax-his2 31172 ax-his3 31173

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-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-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-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-ltxr 11178 df-sub 11373 df-neg 11374 df-hvsub 31060

This theorem is referenced by: his2sub2 31182 hi2eq 31194 pjhthlem1 31480 h1de2i 31642 pjdifnormii 31772 lnopeqi 32097 riesz3i 32151 leop2 32213 hmopidmpji 32241 pjssposi 32261 pjclem4 32288 pj3si 32296

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