hvsubdistr2 - Hilbert Space Explorer

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Theorem hvsubdistr2 30902

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 30865	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2	1	3adant2 1128	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
3		hvmulcl 30865	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
4	3	3adant1 1127	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
5		hvsubval 30868	. . 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 582	. 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 11683	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → (-1 · 𝐵) = -𝐵)
8	7	oveq1d 7430	. . . . . 6 ⊢ (𝐵 ∈ ℂ → ((-1 · 𝐵) ·_ℎ 𝐶) = (-𝐵 ·_ℎ 𝐶))
9	8	adantr 479	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-𝐵 ·_ℎ 𝐶))
10		neg1cn 12354	. . . . . 6 ⊢ -1 ∈ ℂ
11		ax-hvmulass 30859	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
12	10, 11	mp3an1 1444	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
13	9, 12	eqtr3d 2767	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-𝐵 ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
14	13	3adant1 1127	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-𝐵 ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
15	14	oveq2d 7431	. 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 11488	. . . 4 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
17		ax-hvdistr2 30861	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)))
18	16, 17	syl3an2 1161	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)))
19		negsub 11536	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵))
20	19	3adant3 1129	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 + -𝐵) = (𝐴 − 𝐵))
21	20	oveq1d 7430	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 − 𝐵) ·_ℎ 𝐶))
22	18, 21	eqtr3d 2767	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)) = ((𝐴 − 𝐵) ·_ℎ 𝐶))
23	6, 15, 22	3eqtr2rd 2772	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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 (class class class)co 7415 ℂcc 11134 1c1 11137 + caddc 11139 · cmul 11141 − cmin 11472 -cneg 11473 ℋchba 30771 +_ℎ cva 30772 ·_ℎ csm 30773 −_ℎ cmv 30777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-hfvmul 30857 ax-hvmulass 30859 ax-hvdistr2 30861

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-ltxr 11281 df-sub 11474 df-neg 11475 df-hvsub 30823

This theorem is referenced by: hvmulcan2 30925

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