hvsubsub4i - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > hvsubsub4i		Structured version Visualization version GIF version

Theorem hvsubsub4i 30986

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 12352	. . . . . 6 ⊢ -1 ∈ ℂ
3		hvass.2	. . . . . 6 ⊢ 𝐵 ∈ ℋ
4	2, 3	hvmulcli 30941	. . . . 5 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
5		hvass.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
6	2, 5	hvmulcli 30941	. . . . 5 ⊢ (-1 ·_ℎ 𝐶) ∈ ℋ
7		hvadd4.4	. . . . . . 7 ⊢ 𝐷 ∈ ℋ
8	2, 7	hvmulcli 30941	. . . . . 6 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
9	2, 8	hvmulcli 30941	. . . . 5 ⊢ (-1 ·_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
10	1, 4, 6, 9	hvadd4i 30985	. . . 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 30978	. . . . 5 ⊢ (-1 ·_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
12	11	oveq2i 7414	. . . 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 30978	. . . . 5 ⊢ (-1 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
14	13	oveq2i 7414	. . . 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 2768	. . 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 30945	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ∈ ℋ
17	5, 8	hvaddcli 30945	. . . 4 ⊢ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
18	16, 17	hvsubvali 30947	. . 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 30945	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐶)) ∈ ℋ
20	3, 8	hvaddcli 30945	. . . 4 ⊢ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
21	19, 20	hvsubvali 30947	. . 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 2768	. 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 30947	. . 3 ⊢ (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵))
24	5, 7	hvsubvali 30947	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
25	23, 24	oveq12i 7415	. 2 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
26	1, 5	hvsubvali 30947	. . 3 ⊢ (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶))
27	3, 7	hvsubvali 30947	. . 3 ⊢ (𝐵 −_ℎ 𝐷) = (𝐵 +_ℎ (-1 ·_ℎ 𝐷))
28	26, 27	oveq12i 7415	. 2 ⊢ ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) −_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))
29	22, 25, 28	3eqtr4i 2768	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 2108 (class class class)co 7403 1c1 11128 -cneg 11465 ℋchba 30846 +_ℎ cva 30847 ·_ℎ csm 30848 −_ℎ cmv 30852

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-hfvadd 30927 ax-hvcom 30928 ax-hvass 30929 ax-hfvmul 30932 ax-hvdistr1 30935

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11269 df-mnf 11270 df-ltxr 11272 df-sub 11466 df-neg 11467 df-hvsub 30898

This theorem is referenced by: hvsubsub4 30987 pjsslem 31606

W3C validator