Theorem hosass 5607

Description: Associativity of sum of Hilbert space operators.

Hypotheses

Ref	Expression
hods.1	⊢ R: ℋ –→ ℋ
hods.2	⊢ S: ℋ –→ ℋ
hods.3	⊢ T: ℋ –→ ℋ

Assertion

Ref	Expression
hosass	⊢ ((R +P S) +P T) = (R +P (S +P T))

Proof of Theorem hosass

Step	Hyp	Ref	Expression
1		ax-hvass 4986	. . . . 5 ⊢ (((R ‘x) ∈ ℋ ∧ (S ‘x) ∈ ℋ ∧ (T ‘x) ∈ ℋ ) → (((R ‘x) +v (S ‘x)) +v (T ‘x)) = ((R ‘x) +v ((S ‘x) +v (T ‘x))))
2		hods.1	. . . . . 6 ⊢ R: ℋ –→ ℋ
3	2	hocl 5594	. . . . 5 ⊢ (x ∈ ℋ → (R ‘x) ∈ ℋ )
4		hods.2	. . . . . 6 ⊢ S: ℋ –→ ℋ
5	4	hocl 5594	. . . . 5 ⊢ (x ∈ ℋ → (S ‘x) ∈ ℋ )
6		hods.3	. . . . . 6 ⊢ T: ℋ –→ ℋ
7	6	hocl 5594	. . . . 5 ⊢ (x ∈ ℋ → (T ‘x) ∈ ℋ )
8	1, 3, 5, 7	syl3anc 629	. . . 4 ⊢ (x ∈ ℋ → (((R ‘x) +v (S ‘x)) +v (T ‘x)) = ((R ‘x) +v ((S ‘x) +v (T ‘x))))
9	2, 4	hosf 5602	. . . . . . 7 ⊢ (R +P S): ℋ –→ ℋ
10	9, 6	pm3.2i 234	. . . . . 6 ⊢ ((R +P S): ℋ –→ ℋ ∧ T: ℋ –→ ℋ )
11		hosvalt 5489	. . . . . 6 ⊢ ((((R +P S): ℋ –→ ℋ ∧ T: ℋ –→ ℋ ) ∧ x ∈ ℋ ) → (((R +P S) +P T) ‘x) = (((R +P S) ‘x) +v (T ‘x)))
12	10, 11	mpan 518	. . . . 5 ⊢ (x ∈ ℋ → (((R +P S) +P T) ‘x) = (((R +P S) ‘x) +v (T ‘x)))
13	2, 4	pm3.2i 234	. . . . . . 7 ⊢ (R: ℋ –→ ℋ ∧ S: ℋ –→ ℋ )
14		hosvalt 5489	. . . . . . 7 ⊢ (((R: ℋ –→ ℋ ∧ S: ℋ –→ ℋ ) ∧ x ∈ ℋ ) → ((R +P S) ‘x) = ((R ‘x) +v (S ‘x)))
15	13, 14	mpan 518	. . . . . 6 ⊢ (x ∈ ℋ → ((R +P S) ‘x) = ((R ‘x) +v (S ‘x)))
16	15	opreq1d 3012	. . . . 5 ⊢ (x ∈ ℋ → (((R +P S) ‘x) +v (T ‘x)) = (((R ‘x) +v (S ‘x)) +v (T ‘x)))
17	12, 16	eqtrd 1128	. . . 4 ⊢ (x ∈ ℋ → (((R +P S) +P T) ‘x) = (((R ‘x) +v (S ‘x)) +v (T ‘x)))
18	4, 6	hosf 5602	. . . . . . 7 ⊢ (S +P T): ℋ –→ ℋ
19	2, 18	pm3.2i 234	. . . . . 6 ⊢ (R: ℋ –→ ℋ ∧ (S +P T): ℋ –→ ℋ )
20		hosvalt 5489	. . . . . 6 ⊢ (((R: ℋ –→ ℋ ∧ (S +P T): ℋ –→ ℋ ) ∧ x ∈ ℋ ) → ((R +P (S +P T)) ‘x) = ((R ‘x) +v ((S +P T) ‘x)))
21	19, 20	mpan 518	. . . . 5 ⊢ (x ∈ ℋ → ((R +P (S +P T)) ‘x) = ((R ‘x) +v ((S +P T) ‘x)))
22	4, 6	pm3.2i 234	. . . . . . 7 ⊢ (S: ℋ –→ ℋ ∧ T: ℋ –→ ℋ )
23		hosvalt 5489	. . . . . . 7 ⊢ (((S: ℋ –→ ℋ ∧ T: ℋ –→ ℋ ) ∧ x ∈ ℋ ) → ((S +P T) ‘x) = ((S ‘x) +v (T ‘x)))
24	22, 23	mpan 518	. . . . . 6 ⊢ (x ∈ ℋ → ((S +P T) ‘x) = ((S ‘x) +v (T ‘x)))
25	24	opreq2d 3013	. . . . 5 ⊢ (x ∈ ℋ → ((R ‘x) +v ((S +P T) ‘x)) = ((R ‘x) +v ((S ‘x) +v (T ‘x))))
26	21, 25	eqtrd 1128	. . . 4 ⊢ (x ∈ ℋ → ((R +P (S +P T)) ‘x) = ((R ‘x) +v ((S ‘x) +v (T ‘x))))
27	8, 17, 26	3eqtr4d 1134	. . 3 ⊢ (x ∈ ℋ → (((R +P S) +P T) ‘x) = ((R +P (S +P T)) ‘x))
28	27	rgen 1247	. 2 ⊢ ∀x ∈ ℋ (((R +P S) +P T) ‘x) = ((R +P (S +P T)) ‘x)
29	9, 6	hosf 5602	. . 3 ⊢ ((R +P S) +P T): ℋ –→ ℋ
30	2, 18	hosf 5602	. . 3 ⊢ (R +P (S +P T)): ℋ –→ ℋ
31	29, 30	hoeq 5595	. 2 ⊢ (∀x ∈ ℋ (((R +P S) +P T) ‘x) = ((R +P (S +P T)) ‘x) ↔ ((R +P S) +P T) = (R +P (S +P T)))
32	28, 31	mpbi 164	1 ⊢ ((R +P S) +P T) = (R +P (S +P T))

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  –→wf 2418   ‘cfv 2422  (class class class)co 3001   ℋ chil 4958   +_v cva 4959   +_P chos 4977

This theorem is referenced by:  hos12 5608  hosdass 5621

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-hilex 4983  ax-hvaddcl 4984  ax-hvass 4986

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-oprab 3004  df-hosum 5485

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