hoadd32i - Hilbert Space Explorer

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Theorem hoadd32i 32291

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 32285	. . 3 ⊢ (𝑆 +_op 𝑇) = (𝑇 +_op 𝑆)
4	3	oveq2i 7427	. 2 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑅 +_op (𝑇 +_op 𝑆))
5		hods.1	. . 3 ⊢ 𝑅: ℋ⟶ ℋ
6	5, 1, 2	hoaddassi 32289	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	5, 2, 1	hoaddassi 32289	. 2 ⊢ ((𝑅 +_op 𝑇) +_op 𝑆) = (𝑅 +_op (𝑇 +_op 𝑆))
8	4, 6, 7	3eqtr4i 2793	1 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑅 +_op 𝑇) +_op 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ⟶wf 6531 (class class class)co 7416 ℋchba 31432 +_op chos 31451

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-hilex 31512 ax-hfvadd 31513 ax-hvcom 31514 ax-hvass 31515

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7419 df-oprab 7420 df-mpo 7421 df-map 8835 df-hosum 32243

This theorem is used by: hosubeq0i 32339

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