hvsubdistr2 - Hilbert Space Explorer

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Theorem hvsubdistr2 31661

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 31624	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2	1	3adant2 1149	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
3		hvmulcl 31624	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
4	3	3adant1 1148	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
5		hvsubval 31627	. . 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 596	. 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 11764	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → ( -1 · 𝐵) = -𝐵)
8	7	oveq1d 7439	. . . . . 6 ⊢ (𝐵 ∈ ℂ → (( -1 · 𝐵) ·_ℎ 𝐶) = ( -𝐵 ·_ℎ 𝐶))
9	8	adantr 486	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (( -1 · 𝐵) ·_ℎ 𝐶) = ( -𝐵 ·_ℎ 𝐶))
10		neg1cn 12312	. . . . . 6 ⊢ -1 ∈ ℂ
11		ax-hvmulass 31618	. . . . . 6 ⊢ (( -1 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (( -1 · 𝐵) ·_ℎ 𝐶) = ( -1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
12	10, 11	mp3an1 1477	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (( -1 · 𝐵) ·_ℎ 𝐶) = ( -1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
13	9, 12	eqtr3d 2797	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ( -𝐵 ·_ℎ 𝐶) = ( -1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
14	13	3adant1 1148	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ( -𝐵 ·_ℎ 𝐶) = ( -1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
15	14	oveq2d 7440	. 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 11564	. . . 4 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
17		ax-hvdistr2 31620	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ ( -𝐵 ·_ℎ 𝐶)))
18	16, 17	syl3an2 1182	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ ( -𝐵 ·_ℎ 𝐶)))
19		negsub 11613	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵))
20	19	3adant3 1150	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 + -𝐵) = (𝐴 − 𝐵))
21	20	oveq1d 7439	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 − 𝐵) ·_ℎ 𝐶))
22	18, 21	eqtr3d 2797	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) +_ℎ ( -𝐵 ·_ℎ 𝐶)) = ((𝐴 − 𝐵) ·_ℎ 𝐶))
23	6, 15, 22	3eqtr2rd 2802	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 − 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 (class class class)co 7424 ℂcc 11205 1c1 11208 + caddc 11210 · cmul 11212 − cmin 11548 -cneg 11549 ℋchba 31530 +_ℎ cva 31531 ·_ℎ csm 31532 −_ℎ cmv 31536

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-hfvmul 31616 ax-hvmulass 31618 ax-hvdistr2 31620

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8724 df-en 8981 df-dom 8982 df-sdom 8983 df-pnf 11352 df-mnf 11353 df-ltxr 11355 df-sub 11550 df-neg 11551 df-hvsub 31582

This theorem is used by: hvmulcan2 31684

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