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Theorem hvsubass 31703

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 12345	. . . 4 ⊢ -1 ∈ ℂ
2		hvmulcl 31672	. . . 4 ⊢ (( -1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ( -1 ·_ℎ 𝐵) ∈ ℋ)
3	1, 2	mpan 703	. . 3 ⊢ (𝐵 ∈ ℋ → ( -1 ·_ℎ 𝐵) ∈ ℋ)
4		hvaddsubass 31700	. . 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 31675	. . . 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 7466	. 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 31671	. . . . 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 31675	. . . 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 31675	. . . . . . 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 31667	. . . . . . 7 ⊢ (( -1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ( -1 ·_ℎ (𝐵 +_ℎ 𝐶)) = (( -1 ·_ℎ 𝐵) +_ℎ ( -1 ·_ℎ 𝐶)))
18	1, 17	mp3an1 1478	. . . . . 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 2801	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (( -1 ·_ℎ 𝐵) −_ℎ 𝐶) = ( -1 ·_ℎ (𝐵 +_ℎ 𝐶)))
21	20	oveq2d 7467	. . 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 2801	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ (𝐵 +_ℎ 𝐶)) = (𝐴 +_ℎ (( -1 ·_ℎ 𝐵) −_ℎ 𝐶)))
23	5, 8, 22	3eqtr4d 2808	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 2147 (class class class)co 7451 ℂcc 11237 1c1 11240 -cneg 11581 ℋchba 31578 +_ℎ cva 31579 ·_ℎ csm 31580 −_ℎ cmv 31584

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-hfvadd 31659 ax-hvass 31661 ax-hfvmul 31664 ax-hvdistr1 31667

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-po 5566 df-so 5567 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-ltxr 11387 df-sub 11582 df-neg 11583 df-hvsub 31630

This theorem is used by: hvsub32 31704 hvsubassi 31714 pjhthlem1 32050

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