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Theorem hoadd12i 31797

Description: Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (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
hoadd12i	⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑆 +_op (𝑅 +_op 𝑇))

**Proof of Theorem hoadd12i**
Step	Hyp	Ref	Expression
1		hods.1	. . . 4 ⊢ 𝑅: ℋ⟶ ℋ
2		hods.2	. . . 4 ⊢ 𝑆: ℋ⟶ ℋ
3	1, 2	hoaddcomi 31792	. . 3 ⊢ (𝑅 +_op 𝑆) = (𝑆 +_op 𝑅)
4	3	oveq1i 7442	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑆 +_op 𝑅) +_op 𝑇)
5		hods.3	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
6	1, 2, 5	hoaddassi 31796	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	2, 1, 5	hoaddassi 31796	. 2 ⊢ ((𝑆 +_op 𝑅) +_op 𝑇) = (𝑆 +_op (𝑅 +_op 𝑇))
8	4, 6, 7	3eqtr3i 2772	1 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑆 +_op (𝑅 +_op 𝑇))

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ⟶wf 6556 (class class class)co 7432 ℋchba 30939 +_op chos 30958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-hilex 31019 ax-hfvadd 31020 ax-hvcom 31021 ax-hvass 31022

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-hosum 31750

This theorem is referenced by: ho0subi 31815

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