hvaddsub12 - Hilbert Space Explorer

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Theorem hvaddsub12 29508

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 12157	. . . 4 ⊢ -1 ∈ ℂ
2		hvmulcl 29483	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ 𝐶) ∈ ℋ)
3	1, 2	mpan 687	. . 3 ⊢ (𝐶 ∈ ℋ → (-1 ·_ℎ 𝐶) ∈ ℋ)
4		hvadd12 29505	. . 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 1164	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
6		hvsubval 29486	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) = (𝐵 +_ℎ (-1 ·_ℎ 𝐶)))
7	6	oveq2d 7329	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))))
8	7	3adant1 1129	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))))
9		hvsubval 29486	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶)))
10	9	oveq2d 7329	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
11	10	3adant2 1130	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
12	5, 8, 11	3eqtr4d 2787	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 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 (class class class)co 7313 ℂcc 10939 1c1 10942 -cneg 11276 ℋchba 29389 +_ℎ cva 29390 ·_ℎ csm 29391 −_ℎ cmv 29395

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-hvcom 29471 ax-hvass 29472 ax-hfvmul 29475

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4849 df-iun 4937 df-br 5086 df-opab 5148 df-mpt 5169 df-id 5505 df-po 5519 df-so 5520 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-er 8544 df-en 8780 df-dom 8781 df-sdom 8782 df-pnf 11081 df-mnf 11082 df-ltxr 11084 df-sub 11277 df-neg 11278 df-hvsub 29441

This theorem is referenced by: 5oalem1 30124 3oalem2 30133 pjcji 30154 pjclem4 30669 pj3si 30677

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