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Theorem hvsubass 30284

Description: Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

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

**Proof of Theorem hvsubass**
Step	Hyp	Ref	Expression
1		neg1cn 12322	. . . 4 ⊢ -1 ∈ ℂ
2		hvmulcl 30253	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
3	1, 2	mpan 688	. . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
4		hvaddsubass 30281	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ (-1 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ 𝐶) = (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)))
5	3, 4	syl3an2 1164	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ 𝐶) = (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)))
6		hvsubval 30256	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
7	6	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
8	7	oveq1d 7420	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) −_ℎ 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ 𝐶))
9		simp1 1136	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐴 ∈ ℋ)
10		hvaddcl 30252	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
11	10	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
12		hvsubval 30256	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +_ℎ 𝐶) ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ (-1 ·_ℎ (𝐵 +_ℎ 𝐶))))
13	9, 11, 12	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ (-1 ·_ℎ (𝐵 +_ℎ 𝐶))))
14		hvsubval 30256	. . . . . . 7 ⊢ (((-1 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
15	3, 14	sylan 580	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
16	15	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
17		ax-hvdistr1 30248	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
18	1, 17	mp3an1 1448	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
19	18	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
20	16, 19	eqtr4d 2775	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = (-1 ·_ℎ (𝐵 +_ℎ 𝐶)))
21	20	oveq2d 7421	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)) = (𝐴 +_ℎ (-1 ·_ℎ (𝐵 +_ℎ 𝐶))))
22	13, 21	eqtr4d 2775	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)))
23	5, 8, 22	3eqtr4d 2782	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) −_ℎ 𝐶) = (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 (class class class)co 7405 ℂcc 11104 1c1 11107 -cneg 11441 ℋchba 30159 +_ℎ cva 30160 ·_ℎ csm 30161 −_ℎ cmv 30165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-hfvadd 30240 ax-hvass 30242 ax-hfvmul 30245 ax-hvdistr1 30248

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-sub 11442 df-neg 11443 df-hvsub 30211

This theorem is referenced by: hvsub32 30285 hvsubassi 30295 pjhthlem1 30631

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