hoadd12i - Hilbert Space Explorer

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Theorem hoadd12i 32436

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 32431	. . 3 ⊢ (𝑅 +_op 𝑆) = (𝑆 +_op 𝑅)
4	3	oveq1i 7461	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑆 +_op 𝑅) +_op 𝑇)
5		hods.3	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
6	1, 2, 5	hoaddassi 32435	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	2, 1, 5	hoaddassi 32435	. 2 ⊢ ((𝑆 +_op 𝑅) +_op 𝑇) = (𝑆 +_op (𝑅 +_op 𝑇))
8	4, 6, 7	3eqtr3i 2794	1 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑆 +_op (𝑅 +_op 𝑇))

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ⟶wf 6562 (class class class)co 7451 ℋchba 31578 +_op chos 31597

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-hilex 31658 ax-hfvadd 31659 ax-hvcom 31660 ax-hvass 31661

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-map 8887 df-hosum 32389

This theorem is used by: ho0subi 32454

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