Theorem hoaddassi 31720

Description: Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
hoaddassi	⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))

Proof of Theorem hoaddassi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hods.1	. . . . . 6 ⊢ 𝑅: ℋ⟶ ℋ
2		hods.2	. . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ
3		hosval 31684	. . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅‘𝑥) +ℎ (𝑆‘𝑥)))
4	1, 2, 3	mp3an12 1453	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅‘𝑥) +ℎ (𝑆‘𝑥)))
5	4	oveq1d 7364	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)) = (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)))
6	1, 2	hoaddcli 31712	. . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ
7		hods.3	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
8		hosval 31684	. . . . 5 ⊢ (((𝑅 +op 𝑆): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)))
9	6, 7, 8	mp3an12 1453	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)))
10		hosval 31684	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
11	2, 7, 10	mp3an12 1453	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
12	11	oveq2d 7365	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
13	2, 7	hoaddcli 31712	. . . . . 6 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
14		hosval 31684	. . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ (𝑆 +op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)))
15	1, 13, 14	mp3an12 1453	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)))
16	1	ffvelcdmi 7017	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑅‘𝑥) ∈ ℋ)
17	2	ffvelcdmi 7017	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
18	7	ffvelcdmi 7017	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
19		ax-hvass 30946	. . . . . 6 ⊢ (((𝑅‘𝑥) ∈ ℋ ∧ (𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
20	16, 17, 18, 19	syl3anc 1373	. . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
21	12, 15, 20	3eqtr4d 2774	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)))
22	5, 9, 21	3eqtr4d 2774	. . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥))
23	22	rgen 3046	. 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥)
24	6, 7	hoaddcli 31712	. . 3 ⊢ ((𝑅 +op 𝑆) +op 𝑇): ℋ⟶ ℋ
25	1, 13	hoaddcli 31712	. . 3 ⊢ (𝑅 +op (𝑆 +op 𝑇)): ℋ⟶ ℋ
26	24, 25	hoeqi 31705	. 2 ⊢ (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)))
27	23, 26	mpbi 230	1 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ℋchba 30863   +_ℎ cva 30864   +_op chos 30882

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-hilex 30943  ax-hfvadd 30944  ax-hvass 30946

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-map 8755  df-hosum 31674

This theorem is referenced by:  hoadd12i  31721  hoadd32i  31722  hoaddass  31726  hosubeq0i  31770