hoadd12i - Hilbert Space Explorer

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Theorem hoadd12i 31848

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 31843	. . 3 ⊢ (𝑅 +_op 𝑆) = (𝑆 +_op 𝑅)
4	3	oveq1i 7377	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑆 +_op 𝑅) +_op 𝑇)
5		hods.3	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
6	1, 2, 5	hoaddassi 31847	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	2, 1, 5	hoaddassi 31847	. 2 ⊢ ((𝑆 +_op 𝑅) +_op 𝑇) = (𝑆 +_op (𝑅 +_op 𝑇))
8	4, 6, 7	3eqtr3i 2767	1 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑆 +_op (𝑅 +_op 𝑇))

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ⟶wf 6494 (class class class)co 7367 ℋchba 30990 +_op chos 31009

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-hilex 31070 ax-hfvadd 31071 ax-hvcom 31072 ax-hvass 31073

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-map 8775 df-hosum 31801

This theorem is referenced by: ho0subi 31866

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