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Theorem hoadd12i 30040

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 30035	. . 3 ⊢ (𝑅 +_op 𝑆) = (𝑆 +_op 𝑅)
4	3	oveq1i 7265	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑆 +_op 𝑅) +_op 𝑇)
5		hods.3	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
6	1, 2, 5	hoaddassi 30039	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	2, 1, 5	hoaddassi 30039	. 2 ⊢ ((𝑆 +_op 𝑅) +_op 𝑇) = (𝑆 +_op (𝑅 +_op 𝑇))
8	4, 6, 7	3eqtr3i 2774	1 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑆 +_op (𝑅 +_op 𝑇))

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ⟶wf 6414 (class class class)co 7255 ℋchba 29182 +_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 ax-hvass 29265

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: ho0subi 30058

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