hoadd32i - Hilbert Space Explorer

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Theorem hoadd32i 31802

Description: Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (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
hoadd32i	⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑅 +_op 𝑇) +_op 𝑆)

**Proof of Theorem hoadd32i**
Step	Hyp	Ref	Expression
1		hods.2	. . . 4 ⊢ 𝑆: ℋ⟶ ℋ
2		hods.3	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
3	1, 2	hoaddcomi 31796	. . 3 ⊢ (𝑆 +_op 𝑇) = (𝑇 +_op 𝑆)
4	3	oveq2i 7367	. 2 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑅 +_op (𝑇 +_op 𝑆))
5		hods.1	. . 3 ⊢ 𝑅: ℋ⟶ ℋ
6	5, 1, 2	hoaddassi 31800	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	5, 2, 1	hoaddassi 31800	. 2 ⊢ ((𝑅 +_op 𝑇) +_op 𝑆) = (𝑅 +_op (𝑇 +_op 𝑆))
8	4, 6, 7	3eqtr4i 2767	1 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑅 +_op 𝑇) +_op 𝑆)

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ⟶wf 6486 (class class class)co 7356 ℋchba 30943 +_op chos 30962

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 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-hilex 31023 ax-hfvadd 31024 ax-hvcom 31025 ax-hvass 31026

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 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-map 8763 df-hosum 31754

This theorem is referenced by: hosubeq0i 31850

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