his2sub - Hilbert Space Explorer

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Theorem his2sub 31121

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 31045	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
2	1	oveq1d 7446	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
3	2	3adant3 1131	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
4		neg1cn 12378	. . . . 5 ⊢ -1 ∈ ℂ
5		hvmulcl 31042	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
6	4, 5	mpan 690	. . . 4 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
7		ax-his2 31112	. . . 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 1163	. . 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 31113	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
10	4, 9	mp3an1 1447	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
11		hicl 31109	. . . . . . 7 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
12	11	mulm1d 11713	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶))
13	10, 12	eqtrd 2775	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶))
14	13	oveq2d 7447	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
15	14	3adant1 1129	. . 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 2775	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
17		hicl 31109	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
18	17	3adant2 1130	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
19	11	3adant1 1129	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
20	18, 19	negsubd 11624	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))
21	3, 16, 20	3eqtrd 2779	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 1086 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 ℋchba 30948 +_ℎ cva 30949 ·_ℎ csm 30950 ·_ih csp 30951 −_ℎ cmv 30954

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-hfvmul 31034 ax-hfi 31108 ax-his2 31112 ax-his3 31113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-hvsub 31000

This theorem is referenced by: his2sub2 31122 hi2eq 31134 pjhthlem1 31420 h1de2i 31582 pjdifnormii 31712 lnopeqi 32037 riesz3i 32091 leop2 32153 hmopidmpji 32181 pjssposi 32201 pjclem4 32228 pj3si 32236

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