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Theorem hvsubass 31076

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 12407	. . . 4 ⊢ -1 ∈ ℂ
2		hvmulcl 31045	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
3	1, 2	mpan 689	. . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
4		hvaddsubass 31073	. . 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 31048	. . . 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 7463	. 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 31044	. . . . 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 31048	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +_ℎ 𝐶) ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ (-1 ·_ℎ (𝐵 +_ℎ 𝐶))))
13	9, 11, 12	syl2anc 583	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ (-1 ·_ℎ (𝐵 +_ℎ 𝐶))))
14		hvsubval 31048	. . . . . . 7 ⊢ (((-1 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
15	3, 14	sylan 579	. . . . . 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 31040	. . . . . . 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 2783	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = (-1 ·_ℎ (𝐵 +_ℎ 𝐶)))
21	20	oveq2d 7464	. . 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 2783	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)))
23	5, 8, 22	3eqtr4d 2790	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 1537 ∈ wcel 2108 (class class class)co 7448 ℂcc 11182 1c1 11185 -cneg 11521 ℋchba 30951 +_ℎ cva 30952 ·_ℎ csm 30953 −_ℎ cmv 30957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-hfvadd 31032 ax-hvass 31034 ax-hfvmul 31037 ax-hvdistr1 31040

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-ltxr 11329 df-sub 11522 df-neg 11523 df-hvsub 31003

This theorem is referenced by: hvsub32 31077 hvsubassi 31087 pjhthlem1 31423

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