Theorem hoadd32i 31608

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 31602	. . 3 ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
4	3	oveq2i 7437	. 2 ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑅 +op (𝑇 +op 𝑆))
5		hods.1	. . 3 ⊢ 𝑅: ℋ⟶ ℋ
6	5, 1, 2	hoaddassi 31606	. 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
7	5, 2, 1	hoaddassi 31606	. 2 ⊢ ((𝑅 +op 𝑇) +op 𝑆) = (𝑅 +op (𝑇 +op 𝑆))
8	4, 6, 7	3eqtr4i 2766	1 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)

Syntax hints:   = wceq 1533  ⟶wf 6549  (class class class)co 7426   ℋchba 30749   +_op chos 30768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-hilex 30829  ax-hfvadd 30830  ax-hvcom 30831  ax-hvass 30832

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-hosum 31560

This theorem is referenced by:  hosubeq0i  31656