hvsubdistr1 - Hilbert Space Explorer

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Theorem hvsubdistr1 30833

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

**Assertion**
Ref	Expression
hvsubdistr1	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)))

**Proof of Theorem hvsubdistr1**
Step	Hyp	Ref	Expression
1		neg1cn 12342	. . . . 5 ⊢ -1 ∈ ℂ
2		hvmulcl 30797	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ 𝐶) ∈ ℋ)
3	1, 2	mpan 689	. . . 4 ⊢ (𝐶 ∈ ℋ → (-1 ·_ℎ 𝐶) ∈ ℋ)
4		ax-hvdistr1 30792	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ (-1 ·_ℎ 𝐶) ∈ ℋ) → (𝐴 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))) = ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ (-1 ·_ℎ 𝐶))))
5	3, 4	syl3an3 1163	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))) = ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ (-1 ·_ℎ 𝐶))))
6		hvmulcom 30827	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ -1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (-1 ·_ℎ 𝐶)) = (-1 ·_ℎ (𝐴 ·_ℎ 𝐶)))
7	1, 6	mp3an2 1446	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (-1 ·_ℎ 𝐶)) = (-1 ·_ℎ (𝐴 ·_ℎ 𝐶)))
8	7	oveq2d 7430	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ (-1 ·_ℎ 𝐶))) = ((𝐴 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 ·_ℎ 𝐶))))
9	8	3adant2 1129	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ (-1 ·_ℎ 𝐶))) = ((𝐴 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 ·_ℎ 𝐶))))
10	5, 9	eqtrd 2767	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))) = ((𝐴 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 ·_ℎ 𝐶))))
11		hvsubval 30800	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) = (𝐵 +_ℎ (-1 ·_ℎ 𝐶)))
12	11	3adant1 1128	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) = (𝐵 +_ℎ (-1 ·_ℎ 𝐶)))
13	12	oveq2d 7430	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = (𝐴 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))))
14		hvmulcl 30797	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
15	14	3adant3 1130	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
16		hvmulcl 30797	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
17	16	3adant2 1129	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
18		hvsubval 30800	. . 3 ⊢ (((𝐴 ·_ℎ 𝐵) ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 ·_ℎ 𝐶))))
19	15, 17, 18	syl2anc 583	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 ·_ℎ 𝐶))))
20	10, 13, 19	3eqtr4d 2777	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_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 -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-hvdistr1 30792

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: hvsubdistr1i 30836 hvmulcan 30856

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