hvaddsub12 - Hilbert Space Explorer

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Theorem hvaddsub12 30982

Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hvaddsub12	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)))

**Proof of Theorem hvaddsub12**
Step	Hyp	Ref	Expression
1		neg1cn 12113	. . . 4 ⊢ -1 ∈ ℂ
2		hvmulcl 30957	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ 𝐶) ∈ ℋ)
3	1, 2	mpan 690	. . 3 ⊢ (𝐶 ∈ ℋ → (-1 ·_ℎ 𝐶) ∈ ℋ)
4		hvadd12 30979	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (-1 ·_ℎ 𝐶) ∈ ℋ) → (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
5	3, 4	syl3an3 1165	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
6		hvsubval 30960	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) = (𝐵 +_ℎ (-1 ·_ℎ 𝐶)))
7	6	oveq2d 7365	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))))
8	7	3adant1 1130	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))))
9		hvsubval 30960	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶)))
10	9	oveq2d 7365	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
11	10	3adant2 1131	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
12	5, 8, 11	3eqtr4d 2774	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 (class class class)co 7349 ℂcc 11007 1c1 11010 -cneg 11348 ℋchba 30863 +_ℎ cva 30864 ·_ℎ csm 30865 −_ℎ cmv 30869

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-hvcom 30945 ax-hvass 30946 ax-hfvmul 30949

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-ltxr 11154 df-sub 11349 df-neg 11350 df-hvsub 30915

This theorem is referenced by: 5oalem1 31598 3oalem2 31607 pjcji 31628 pjclem4 32143 pj3si 32151

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