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Theorem hvsubass 29406

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 12087	. . . 4 ⊢ -1 ∈ ℂ
2		hvmulcl 29375	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
3	1, 2	mpan 687	. . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
4		hvaddsubass 29403	. . 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 1163	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ 𝐶) = (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)))
6		hvsubval 29378	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
7	6	3adant3 1131	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
8	7	oveq1d 7290	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) −_ℎ 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ 𝐶))
9		simp1 1135	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐴 ∈ ℋ)
10		hvaddcl 29374	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
11	10	3adant1 1129	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
12		hvsubval 29378	. . . 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 29378	. . . . . . 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 1129	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
17		ax-hvdistr1 29370	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
18	1, 17	mp3an1 1447	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
19	18	3adant1 1129	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
20	16, 19	eqtr4d 2781	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = (-1 ·_ℎ (𝐵 +_ℎ 𝐶)))
21	20	oveq2d 7291	. . 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 2781	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)))
23	5, 8, 22	3eqtr4d 2788	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 1086 = wceq 1539 ∈ wcel 2106 (class class class)co 7275 ℂcc 10869 1c1 10872 -cneg 11206 ℋchba 29281 +_ℎ cva 29282 ·_ℎ csm 29283 −_ℎ cmv 29287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-hfvadd 29362 ax-hvass 29364 ax-hfvmul 29367 ax-hvdistr1 29370

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 df-sub 11207 df-neg 11208 df-hvsub 29333

This theorem is referenced by: hvsub32 29407 hvsubassi 29417 pjhthlem1 29753

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