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Theorem hoadd32i 22358
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
StepHypRef Expression
1 hods.2 . . . 4  |-  S : ~H
--> ~H
2 hods.3 . . . 4  |-  T : ~H
--> ~H
31, 2hoaddcomi 22352 . . 3  |-  ( S 
+op  T )  =  ( T  +op  S
)
43oveq2i 5869 . 2  |-  ( R 
+op  ( S  +op  T ) )  =  ( R  +op  ( T 
+op  S ) )
5 hods.1 . . 3  |-  R : ~H
--> ~H
65, 1, 2hoaddassi 22356 . 2  |-  ( ( R  +op  S ) 
+op  T )  =  ( R  +op  ( S  +op  T ) )
75, 2, 1hoaddassi 22356 . 2  |-  ( ( R  +op  T ) 
+op  S )  =  ( R  +op  ( T  +op  S ) )
84, 6, 73eqtr4i 2313 1  |-  ( ( R  +op  S ) 
+op  T )  =  ( ( R  +op  T )  +op  S )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   -->wf 5251  (class class class)co 5858   ~Hchil 21499    +op chos 21518
This theorem is referenced by:  hosubeq0i  22406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-hosum 22310
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