hoadd32i - Hilbert Space Explorer

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Theorem hoadd32i 29549

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 29543	. . 3 ⊢ (𝑆 +_op 𝑇) = (𝑇 +_op 𝑆)
4	3	oveq2i 7161	. 2 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑅 +_op (𝑇 +_op 𝑆))
5		hods.1	. . 3 ⊢ 𝑅: ℋ⟶ ℋ
6	5, 1, 2	hoaddassi 29547	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	5, 2, 1	hoaddassi 29547	. 2 ⊢ ((𝑅 +_op 𝑇) +_op 𝑆) = (𝑅 +_op (𝑇 +_op 𝑆))
8	4, 6, 7	3eqtr4i 2854	1 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑅 +_op 𝑇) +_op 𝑆)

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ⟶wf 6345 (class class class)co 7150 ℋchba 28690 +_op chos 28709

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-hosum 29501

This theorem is referenced by: hosubeq0i 29597

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