hvsubsub4i - Hilbert Space Explorer

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Theorem hvsubsub4i 31045

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 12359	. . . . . 6 ⊢ -1 ∈ ℂ
3		hvass.2	. . . . . 6 ⊢ 𝐵 ∈ ℋ
4	2, 3	hvmulcli 31000	. . . . 5 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
5		hvass.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
6	2, 5	hvmulcli 31000	. . . . 5 ⊢ (-1 ·_ℎ 𝐶) ∈ ℋ
7		hvadd4.4	. . . . . . 7 ⊢ 𝐷 ∈ ℋ
8	2, 7	hvmulcli 31000	. . . . . 6 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
9	2, 8	hvmulcli 31000	. . . . 5 ⊢ (-1 ·_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
10	1, 4, 6, 9	hvadd4i 31044	. . . 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 31037	. . . . 5 ⊢ (-1 ·_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
12	11	oveq2i 7421	. . . 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 31037	. . . . 5 ⊢ (-1 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
14	13	oveq2i 7421	. . . 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 2769	. . 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 31004	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ∈ ℋ
17	5, 8	hvaddcli 31004	. . . 4 ⊢ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
18	16, 17	hvsubvali 31006	. . 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 31004	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐶)) ∈ ℋ
20	3, 8	hvaddcli 31004	. . . 4 ⊢ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
21	19, 20	hvsubvali 31006	. . 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 2769	. 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 31006	. . 3 ⊢ (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵))
24	5, 7	hvsubvali 31006	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
25	23, 24	oveq12i 7422	. 2 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
26	1, 5	hvsubvali 31006	. . 3 ⊢ (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶))
27	3, 7	hvsubvali 31006	. . 3 ⊢ (𝐵 −_ℎ 𝐷) = (𝐵 +_ℎ (-1 ·_ℎ 𝐷))
28	26, 27	oveq12i 7422	. 2 ⊢ ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) −_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))
29	22, 25, 28	3eqtr4i 2769	1 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷))

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 (class class class)co 7410 1c1 11135 -cneg 11472 ℋchba 30905 +_ℎ cva 30906 ·_ℎ csm 30907 −_ℎ cmv 30911

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 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-hfvadd 30986 ax-hvcom 30987 ax-hvass 30988 ax-hfvmul 30991 ax-hvdistr1 30994

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-po 5566 df-so 5567 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-ltxr 11279 df-sub 11473 df-neg 11474 df-hvsub 30957

This theorem is referenced by: hvsubsub4 31046 pjsslem 31665

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