Theorem hoaddassi 31799

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 31763	. . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅‘𝑥) +ℎ (𝑆‘𝑥)))
4	1, 2, 3	mp3an12 1451	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅‘𝑥) +ℎ (𝑆‘𝑥)))
5	4	oveq1d 7460	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)) = (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)))
6	1, 2	hoaddcli 31791	. . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ
7		hods.3	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
8		hosval 31763	. . . . 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 1451	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)))
10		hosval 31763	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
11	2, 7, 10	mp3an12 1451	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
12	11	oveq2d 7461	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
13	2, 7	hoaddcli 31791	. . . . . 6 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
14		hosval 31763	. . . . . 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 1451	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)))
16	1	ffvelcdmi 7115	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑅‘𝑥) ∈ ℋ)
17	2	ffvelcdmi 7115	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
18	7	ffvelcdmi 7115	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
19		ax-hvass 31025	. . . . . 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 1371	. . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
21	12, 15, 20	3eqtr4d 2784	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)))
22	5, 9, 21	3eqtr4d 2784	. . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥))
23	22	rgen 3065	. 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥)
24	6, 7	hoaddcli 31791	. . 3 ⊢ ((𝑅 +op 𝑆) +op 𝑇): ℋ⟶ ℋ
25	1, 13	hoaddcli 31791	. . 3 ⊢ (𝑅 +op (𝑆 +op 𝑇)): ℋ⟶ ℋ
26	24, 25	hoeqi 31784	. 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 1537   ∈ wcel 2103  ∀wral 3063  ⟶wf 6568  ‘cfv 6572  (class class class)co 7445   ℋchba 30942   +_ℎ cva 30943   +_op chos 30961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-hilex 31022  ax-hfvadd 31023  ax-hvass 31025

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-map 8882  df-hosum 31753

This theorem is referenced by:  hoadd12i  31800  hoadd32i  31801  hoaddass  31805  hosubeq0i  31849