Theorem hoadd32i 28637

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 28631	. . 3 |- ( S +op T ) = ( T +op S )
4	3	oveq2i 6661	. 2 |- ( R +op ( S +op T ) ) = ( R +op ( T +op S ) )
5		hods.1	. . 3 |- R : ~H --> ~H
6	5, 1, 2	hoaddassi 28635	. 2 |- ( ( R +op S ) +op T ) = ( R +op ( S +op T ) )
7	5, 2, 1	hoaddassi 28635	. 2 |- ( ( R +op T ) +op S ) = ( R +op ( T +op S ) )
8	4, 6, 7	3eqtr4i 2654	1 |- ( ( R +op S ) +op T ) = ( ( R +op T ) +op S )

Syntax hints:   = wceq 1483   -->wf 5884  (class class class)co 6650   ~Hchil 27776   +op chos 27795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-hosum 28589

This theorem is referenced by:  hosubeq0i  28685