Theorem hoaddassi 32405

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 32369	. . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅‘𝑥) +ℎ (𝑆‘𝑥)))
4	1, 2, 3	mp3an12 1481	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅‘𝑥) +ℎ (𝑆‘𝑥)))
5	4	oveq1d 7455	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)) = (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)))
6	1, 2	hoaddcli 32397	. . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ
7		hods.3	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
8		hosval 32369	. . . . 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 1481	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)))
10		hosval 32369	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
11	2, 7, 10	mp3an12 1481	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
12	11	oveq2d 7456	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
13	2, 7	hoaddcli 32397	. . . . . 6 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
14		hosval 32369	. . . . . 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 1481	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)))
16	1	ffvelcdmi 7104	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑅‘𝑥) ∈ ℋ)
17	2	ffvelcdmi 7104	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
18	7	ffvelcdmi 7104	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
19		ax-hvass 31631	. . . . . 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 1399	. . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
21	12, 15, 20	3eqtr4d 2808	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)))
22	5, 9, 21	3eqtr4d 2808	. . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥))
23	22	rgen 3082	. 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥)
24	6, 7	hoaddcli 32397	. . 3 ⊢ ((𝑅 +op 𝑆) +op 𝑇): ℋ⟶ ℋ
25	1, 13	hoaddcli 32397	. . 3 ⊢ (𝑅 +op (𝑆 +op 𝑇)): ℋ⟶ ℋ
26	24, 25	hoeqi 32390	. 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 233	1 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ℋchba 31548   +_ℎ cva 31549   +_op chos 31567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-hilex 31628  ax-hfvadd 31629  ax-hvass 31631

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-map 8874  df-hosum 32359

This theorem is used by:  hoadd12i  32406  hoadd32i  32407  hoaddass  32411  hosubeq0i  32455