hoadd32i - Hilbert Space Explorer

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Theorem hoadd32i 31868

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 31862	. . 3 ⊢ (𝑆 +_op 𝑇) = (𝑇 +_op 𝑆)
4	3	oveq2i 7373	. 2 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑅 +_op (𝑇 +_op 𝑆))
5		hods.1	. . 3 ⊢ 𝑅: ℋ⟶ ℋ
6	5, 1, 2	hoaddassi 31866	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	5, 2, 1	hoaddassi 31866	. 2 ⊢ ((𝑅 +_op 𝑇) +_op 𝑆) = (𝑅 +_op (𝑇 +_op 𝑆))
8	4, 6, 7	3eqtr4i 2770	1 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑅 +_op 𝑇) +_op 𝑆)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ⟶wf 6490 (class class class)co 7362 ℋchba 31009 +_op chos 31028

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 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-hilex 31089 ax-hfvadd 31090 ax-hvcom 31091 ax-hvass 31092

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5521 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8770 df-hosum 31820

This theorem is referenced by: hosubeq0i 31916

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