hoadd32i - Hilbert Space Explorer

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Theorem hoadd32i 31871

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 31865	. . 3 ⊢ (𝑆 +_op 𝑇) = (𝑇 +_op 𝑆)
4	3	oveq2i 7371	. 2 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑅 +_op (𝑇 +_op 𝑆))
5		hods.1	. . 3 ⊢ 𝑅: ℋ⟶ ℋ
6	5, 1, 2	hoaddassi 31869	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	5, 2, 1	hoaddassi 31869	. 2 ⊢ ((𝑅 +_op 𝑇) +_op 𝑆) = (𝑅 +_op (𝑇 +_op 𝑆))
8	4, 6, 7	3eqtr4i 2774	1 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑅 +_op 𝑇) +_op 𝑆)

Colors of variables: wff setvar class

Syntax hints: = wceq 1548 ⟶wf 6485 (class class class)co 7360 ℋchba 31012 +_op chos 31031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-hilex 31092 ax-hfvadd 31093 ax-hvcom 31094 ax-hvass 31095

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7363 df-oprab 7364 df-mpo 7365 df-map 8769 df-hosum 31823

This theorem is referenced by: hosubeq0i 31919

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