hvsubsub4i - Hilbert Space Explorer

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Theorem hvsubsub4i 31083

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 12128	. . . . . 6 ⊢ -1 ∈ ℂ
3		hvass.2	. . . . . 6 ⊢ 𝐵 ∈ ℋ
4	2, 3	hvmulcli 31038	. . . . 5 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
5		hvass.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
6	2, 5	hvmulcli 31038	. . . . 5 ⊢ (-1 ·_ℎ 𝐶) ∈ ℋ
7		hvadd4.4	. . . . . . 7 ⊢ 𝐷 ∈ ℋ
8	2, 7	hvmulcli 31038	. . . . . 6 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
9	2, 8	hvmulcli 31038	. . . . 5 ⊢ (-1 ·_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
10	1, 4, 6, 9	hvadd4i 31082	. . . 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 31075	. . . . 5 ⊢ (-1 ·_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
12	11	oveq2i 7367	. . . 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 31075	. . . . 5 ⊢ (-1 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
14	13	oveq2i 7367	. . . 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 2767	. . 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 31042	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ∈ ℋ
17	5, 8	hvaddcli 31042	. . . 4 ⊢ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
18	16, 17	hvsubvali 31044	. . 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 31042	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐶)) ∈ ℋ
20	3, 8	hvaddcli 31042	. . . 4 ⊢ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
21	19, 20	hvsubvali 31044	. . 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 2767	. 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 31044	. . 3 ⊢ (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵))
24	5, 7	hvsubvali 31044	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
25	23, 24	oveq12i 7368	. 2 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
26	1, 5	hvsubvali 31044	. . 3 ⊢ (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶))
27	3, 7	hvsubvali 31044	. . 3 ⊢ (𝐵 −_ℎ 𝐷) = (𝐵 +_ℎ (-1 ·_ℎ 𝐷))
28	26, 27	oveq12i 7368	. 2 ⊢ ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) −_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))
29	22, 25, 28	3eqtr4i 2767	1 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷))

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 (class class class)co 7356 1c1 11025 -cneg 11363 ℋchba 30943 +_ℎ cva 30944 ·_ℎ csm 30945 −_ℎ cmv 30949

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-hfvadd 31024 ax-hvcom 31025 ax-hvass 31026 ax-hfvmul 31029 ax-hvdistr1 31032

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-po 5530 df-so 5531 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-ltxr 11169 df-sub 11364 df-neg 11365 df-hvsub 30995

This theorem is referenced by: hvsubsub4 31084 pjsslem 31703

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