Theorem hoaddcomi 31754

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	ffvelcdmi 7022	. . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
3		hoeq.2	. . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ
4	3	ffvelcdmi 7022	. . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
5		ax-hvcom 30983	. . . . 5 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥)))
6	2, 4, 5	syl2anc 584	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥)))
7		hosval 31722	. . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
8	1, 3, 7	mp3an12 1453	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
9		hosval 31722	. . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥)))
10	3, 1, 9	mp3an12 1453	. . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥)))
11	6, 8, 10	3eqtr4d 2778	. . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥))
12	11	rgen 3050	. 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)
13	1, 3	hoaddcli 31750	. . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
14	3, 1	hoaddcli 31750	. . 3 ⊢ (𝑇 +op 𝑆): ℋ⟶ ℋ
15	13, 14	hoeqi 31743	. 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
16	12, 15	mpbi 230	1 ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆)

Syntax hints:   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ℋchba 30901   +_ℎ cva 30902   +_op chos 30920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-hilex 30981  ax-hfvadd 30982  ax-hvcom 30983

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8758  df-hosum 31712

This theorem is referenced by:  hoaddcom  31756  hoadd12i  31759  hoadd32i  31760  hoaddsubi  31803  hosd1i  31804  hosubeq0i  31808