hvsubsub4i - Hilbert Space Explorer

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Theorem hvsubsub4i 28942

Description: Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hvass.1	⊢ 𝐴 ∈ ℋ
hvass.2	⊢ 𝐵 ∈ ℋ
hvass.3	⊢ 𝐶 ∈ ℋ
hvadd4.4	⊢ 𝐷 ∈ ℋ

**Assertion**
Ref	Expression
hvsubsub4i	⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷))

**Proof of Theorem hvsubsub4i**
Step	Hyp	Ref	Expression
1		hvass.1	. . . . 5 ⊢ 𝐴 ∈ ℋ
2		neg1cn 11789	. . . . . 6 ⊢ -1 ∈ ℂ
3		hvass.2	. . . . . 6 ⊢ 𝐵 ∈ ℋ
4	2, 3	hvmulcli 28897	. . . . 5 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
5		hvass.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
6	2, 5	hvmulcli 28897	. . . . 5 ⊢ (-1 ·_ℎ 𝐶) ∈ ℋ
7		hvadd4.4	. . . . . . 7 ⊢ 𝐷 ∈ ℋ
8	2, 7	hvmulcli 28897	. . . . . 6 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
9	2, 8	hvmulcli 28897	. . . . 5 ⊢ (-1 ·_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
10	1, 4, 6, 9	hvadd4i 28941	. . . 4 ⊢ ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) +_ℎ ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷))))
11	2, 5, 8	hvdistr1i 28934	. . . . 5 ⊢ (-1 ·_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
12	11	oveq2i 7162	. . . 4 ⊢ ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) +_ℎ (-1 ·_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) +_ℎ ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷))))
13	2, 3, 8	hvdistr1i 28934	. . . . 5 ⊢ (-1 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
14	13	oveq2i 7162	. . . 4 ⊢ ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ (-1 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷))))
15	10, 12, 14	3eqtr4i 2792	. . 3 ⊢ ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) +_ℎ (-1 ·_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ (-1 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))))
16	1, 4	hvaddcli 28901	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ∈ ℋ
17	5, 8	hvaddcli 28901	. . . 4 ⊢ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
18	16, 17	hvsubvali 28903	. . 3 ⊢ ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷))) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) +_ℎ (-1 ·_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷))))
19	1, 6	hvaddcli 28901	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐶)) ∈ ℋ
20	3, 8	hvaddcli 28901	. . . 4 ⊢ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
21	19, 20	hvsubvali 28903	. . 3 ⊢ ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) −_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ (-1 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))))
22	15, 18, 21	3eqtr4i 2792	. 2 ⊢ ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷))) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) −_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))
23	1, 3	hvsubvali 28903	. . 3 ⊢ (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵))
24	5, 7	hvsubvali 28903	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
25	23, 24	oveq12i 7163	. 2 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
26	1, 5	hvsubvali 28903	. . 3 ⊢ (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶))
27	3, 7	hvsubvali 28903	. . 3 ⊢ (𝐵 −_ℎ 𝐷) = (𝐵 +_ℎ (-1 ·_ℎ 𝐷))
28	26, 27	oveq12i 7163	. 2 ⊢ ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) −_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))
29	22, 25, 28	3eqtr4i 2792	1 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷))

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2112 (class class class)co 7151 1c1 10577 -cneg 10910 ℋchba 28802 +_ℎ cva 28803 ·_ℎ csm 28804 −_ℎ cmv 28808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-resscn 10633 ax-1cn 10634 ax-icn 10635 ax-addcl 10636 ax-addrcl 10637 ax-mulcl 10638 ax-mulrcl 10639 ax-mulcom 10640 ax-addass 10641 ax-mulass 10642 ax-distr 10643 ax-i2m1 10644 ax-1ne0 10645 ax-1rid 10646 ax-rnegex 10647 ax-rrecex 10648 ax-cnre 10649 ax-pre-lttri 10650 ax-pre-lttrn 10651 ax-pre-ltadd 10652 ax-hfvadd 28883 ax-hvcom 28884 ax-hvass 28885 ax-hfvmul 28888 ax-hvdistr1 28891

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-op 4530 df-uni 4800 df-iun 4886 df-br 5034 df-opab 5096 df-mpt 5114 df-id 5431 df-po 5444 df-so 5445 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-er 8300 df-en 8529 df-dom 8530 df-sdom 8531 df-pnf 10716 df-mnf 10717 df-ltxr 10719 df-sub 10911 df-neg 10912 df-hvsub 28854

This theorem is referenced by: hvsubsub4 28943 pjsslem 29562

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