hvsubdistr2 - Hilbert Space Explorer

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Theorem hvsubdistr2 30834

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 30797	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2	1	3adant2 1129	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
3		hvmulcl 30797	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
4	3	3adant1 1128	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
5		hvsubval 30800	. . 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 583	. 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 11671	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → (-1 · 𝐵) = -𝐵)
8	7	oveq1d 7429	. . . . . 6 ⊢ (𝐵 ∈ ℂ → ((-1 · 𝐵) ·_ℎ 𝐶) = (-𝐵 ·_ℎ 𝐶))
9	8	adantr 480	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-𝐵 ·_ℎ 𝐶))
10		neg1cn 12342	. . . . . 6 ⊢ -1 ∈ ℂ
11		ax-hvmulass 30791	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
12	10, 11	mp3an1 1445	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((-1 · 𝐵) ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
13	9, 12	eqtr3d 2769	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-𝐵 ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
14	13	3adant1 1128	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-𝐵 ·_ℎ 𝐶) = (-1 ·_ℎ (𝐵 ·_ℎ 𝐶)))
15	14	oveq2d 7430	. 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 11476	. . . 4 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
17		ax-hvdistr2 30793	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)))
18	16, 17	syl3an2 1162	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)))
19		negsub 11524	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵))
20	19	3adant3 1130	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 + -𝐵) = (𝐴 − 𝐵))
21	20	oveq1d 7429	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + -𝐵) ·_ℎ 𝐶) = ((𝐴 − 𝐵) ·_ℎ 𝐶))
22	18, 21	eqtr3d 2769	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) +_ℎ (-𝐵 ·_ℎ 𝐶)) = ((𝐴 − 𝐵) ·_ℎ 𝐶))
23	6, 15, 22	3eqtr2rd 2774	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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 (class class class)co 7414 ℂcc 11122 1c1 11125 + caddc 11127 · cmul 11129 − cmin 11460 -cneg 11461 ℋchba 30703 +_ℎ cva 30704 ·_ℎ csm 30705 −_ℎ cmv 30709

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-hfvmul 30789 ax-hvmulass 30791 ax-hvdistr2 30793

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 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 5143 df-opab 5205 df-mpt 5226 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-ltxr 11269 df-sub 11462 df-neg 11463 df-hvsub 30755

This theorem is referenced by: hvmulcan2 30857

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