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Theorem hvsubass 31526

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 12228	. . . 4 ⊢ -1 ∈ ℂ
2		hvmulcl 31495	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
3	1, 2	mpan 703	. . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
4		hvaddsubass 31523	. . 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 1182	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ 𝐶) = (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)))
6		hvsubval 31498	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
7	6	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
8	7	oveq1d 7429	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) −_ℎ 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ 𝐶))
9		simp1 1154	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐴 ∈ ℋ)
10		hvaddcl 31494	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
11	10	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
12		hvsubval 31498	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +_ℎ 𝐶) ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ (-1 ·_ℎ (𝐵 +_ℎ 𝐶))))
13	9, 11, 12	syl2anc 596	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ (-1 ·_ℎ (𝐵 +_ℎ 𝐶))))
14		hvsubval 31498	. . . . . . 7 ⊢ (((-1 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
15	3, 14	sylan 592	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
16	15	3adant1 1148	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
17		ax-hvdistr1 31490	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
18	1, 17	mp3an1 1477	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
19	18	3adant1 1148	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ 𝐶)))
20	16, 19	eqtr4d 2798	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) −_ℎ 𝐶) = (-1 ·_ℎ (𝐵 +_ℎ 𝐶)))
21	20	oveq2d 7430	. . 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 2798	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ ((-1 ·_ℎ 𝐵) −_ℎ 𝐶)))
23	5, 8, 22	3eqtr4d 2805	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) −_ℎ 𝐶) = (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 (class class class)co 7414 ℂcc 11123 1c1 11126 -cneg 11467 ℋchba 31401 +_ℎ cva 31402 ·_ℎ csm 31403 −_ℎ cmv 31407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-hfvadd 31482 ax-hvass 31484 ax-hfvmul 31487 ax-hvdistr1 31490

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8697 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11270 df-mnf 11271 df-ltxr 11273 df-sub 11468 df-neg 11469 df-hvsub 31453

This theorem is used by: hvsub32 31527 hvsubassi 31537 pjhthlem1 31873

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