his2sub - Hilbert Space Explorer

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Theorem his2sub 30131

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 30055	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
2	1	oveq1d 7392	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
3	2	3adant3 1132	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
4		neg1cn 12291	. . . . 5 ⊢ -1 ∈ ℂ
5		hvmulcl 30052	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
6	4, 5	mpan 688	. . . 4 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
7		ax-his2 30122	. . . 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 1164	. . 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 30123	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
10	4, 9	mp3an1 1448	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
11		hicl 30119	. . . . . . 7 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
12	11	mulm1d 11631	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶))
13	10, 12	eqtrd 2771	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶))
14	13	oveq2d 7393	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
15	14	3adant1 1130	. . 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 2771	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
17		hicl 30119	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
18	17	3adant2 1131	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
19	11	3adant1 1130	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
20	18, 19	negsubd 11542	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))
21	3, 16, 20	3eqtrd 2775	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 (class class class)co 7377 ℂcc 11073 1c1 11076 + caddc 11078 · cmul 11080 − cmin 11409 -cneg 11410 ℋchba 29958 +_ℎ cva 29959 ·_ℎ csm 29960 ·_ih csp 29961 −_ℎ cmv 29964

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-hfvmul 30044 ax-hfi 30118 ax-his2 30122 ax-his3 30123

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-po 5565 df-so 5566 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-pnf 11215 df-mnf 11216 df-ltxr 11218 df-sub 11411 df-neg 11412 df-hvsub 30010

This theorem is referenced by: his2sub2 30132 hi2eq 30144 pjhthlem1 30430 h1de2i 30592 pjdifnormii 30722 lnopeqi 31047 riesz3i 31101 leop2 31163 hmopidmpji 31191 pjssposi 31211 pjclem4 31238 pj3si 31246

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