hvsubdistr2 - Hilbert Space Explorer

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Theorem hvsubdistr2 30303

Description: Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hvsubdistr2	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 − 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)))

**Proof of Theorem hvsubdistr2**
Step	Hyp	Ref	Expression
1		hvmulcl 30266	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2	1	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
3		hvmulcl 30266	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
4	3	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
5		hvsubval 30269	. . 3 ⊢ (((𝐴 ·_ℎ 𝐶) ∈ ℋ ∧ (𝐵 ·_ℎ 𝐶) ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (𝐵 ·_ℎ 𝐶))))
6	2, 4, 5	syl2anc 585	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (𝐵 ·_ℎ 𝐶))))
7		mulm1 11655	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → (-1 · 𝐵) = -𝐵)
8	7	oveq1d 7424	. . . . . 6 ⊢ (𝐵 ∈ ℂ → ((-1 · 𝐵) ·_ℎ 𝐶) = (-𝐵 ·_ℎ 𝐶))
9	8	adantr 482	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-𝐵 ·_ℎ 𝐶))
10		neg1cn 12326	. . . . . 6 ⊢ -1 ∈ ℂ
11		ax-hvmulass 30260	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
12	10, 11	mp3an1 1449	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
13	9, 12	eqtr3d 2775	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-𝐵 ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
14	13	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-𝐵 ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
15	14	oveq2d 7425	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (𝐵 ·_ℎ 𝐶))))
16		negcl 11460	. . . 4 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
17		ax-hvdistr2 30262	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)))
18	16, 17	syl3an2 1165	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)))
19		negsub 11508	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵))
20	19	3adant3 1133	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 + -𝐵) = (𝐴 − 𝐵))
21	20	oveq1d 7424	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 − 𝐵) ·_ℎ 𝐶))
22	18, 21	eqtr3d 2775	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)) = ((𝐴 − 𝐵) ·_ℎ 𝐶))
23	6, 15, 22	3eqtr2rd 2780	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 − 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 (class class class)co 7409 ℂcc 11108 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11444 -cneg 11445 ℋchba 30172 +_ℎ cva 30173 ·_ℎ csm 30174 −_ℎ cmv 30178

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-hfvmul 30258 ax-hvmulass 30260 ax-hvdistr2 30262

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-neg 11447 df-hvsub 30224

This theorem is referenced by: hvmulcan2 30326

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