Theorem hoaddassi 32289

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 32253	. . . . . 6 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅‘𝑥) +ℎ (𝑆‘𝑥)))
4	1, 2, 3	mp3an12 1480	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅‘𝑥) +ℎ (𝑆‘𝑥)))
5	4	oveq1d 7431	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)) = (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)))
6	1, 2	hoaddcli 32281	. . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ
7		hods.3	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
8		hosval 32253	. . . . 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 1480	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) +ℎ (𝑇‘𝑥)))
10		hosval 32253	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
11	2, 7, 10	mp3an12 1480	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
12	11	oveq2d 7432	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
13	2, 7	hoaddcli 32281	. . . . . 6 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
14		hosval 32253	. . . . . 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 1480	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅‘𝑥) +ℎ ((𝑆 +op 𝑇)‘𝑥)))
16	1	ffvelcdmi 7079	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑅‘𝑥) ∈ ℋ)
17	2	ffvelcdmi 7079	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
18	7	ffvelcdmi 7079	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
19		ax-hvass 31515	. . . . . 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 1398	. . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)) = ((𝑅‘𝑥) +ℎ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
21	12, 15, 20	3eqtr4d 2805	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = (((𝑅‘𝑥) +ℎ (𝑆‘𝑥)) +ℎ (𝑇‘𝑥)))
22	5, 9, 21	3eqtr4d 2805	. . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥))
23	22	rgen 3078	. 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥)
24	6, 7	hoaddcli 32281	. . 3 ⊢ ((𝑅 +op 𝑆) +op 𝑇): ℋ⟶ ℋ
25	1, 13	hoaddcli 32281	. . 3 ⊢ (𝑅 +op (𝑆 +op 𝑇)): ℋ⟶ ℋ
26	24, 25	hoeqi 32274	. 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 2145  ∀wral 3076  ⟶wf 6531  ‘cfv 6535  (class class class)co 7416   ℋchba 31432   +_ℎ cva 31433   +_op chos 31451

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-hilex 31512  ax-hfvadd 31513  ax-hvass 31515

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8835  df-hosum 32243

This theorem is used by:  hoadd12i  32290  hoadd32i  32291  hoaddass  32295  hosubeq0i  32339