Theorem hoaddcomi 30035

Description: Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hoeq.1	⊢ 𝑆: ℋ⟶ ℋ
hoeq.2	⊢ 𝑇: ℋ⟶ ℋ

Assertion

Ref	Expression
hoaddcomi	⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆)

Proof of Theorem hoaddcomi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoeq.1	. . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ
2	1	ffvelrni 6942	. . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
3		hoeq.2	. . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ
4	3	ffvelrni 6942	. . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
5		ax-hvcom 29264	. . . . 5 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥)))
6	2, 4, 5	syl2anc 583	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥)))
7		hosval 30003	. . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
8	1, 3, 7	mp3an12 1449	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
9		hosval 30003	. . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥)))
10	3, 1, 9	mp3an12 1449	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥)))
11	6, 8, 10	3eqtr4d 2788	. . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥))
12	11	rgen 3073	. 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)
13	1, 3	hoaddcli 30031	. . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
14	3, 1	hoaddcli 30031	. . 3 ⊢ (𝑇 +op 𝑆): ℋ⟶ ℋ
15	13, 14	hoeqi 30024	. 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
16	12, 15	mpbi 229	1 ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆)

Syntax hints:   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ℋchba 29182   +_ℎ cva 29183   +_op chos 29201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-hilex 29262  ax-hfvadd 29263  ax-hvcom 29264

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-hosum 29993

This theorem is referenced by:  hoaddcom  30037  hoadd12i  30040  hoadd32i  30041  hoaddsubi  30084  hosd1i  30085  hosubeq0i  30089