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Theorem hoadd12i 23237
Description: Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (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
hoadd12i  |-  ( R 
+op  ( S  +op  T ) )  =  ( S  +op  ( R 
+op  T ) )

Proof of Theorem hoadd12i
StepHypRef Expression
1 hods.1 . . . 4  |-  R : ~H
--> ~H
2 hods.2 . . . 4  |-  S : ~H
--> ~H
31, 2hoaddcomi 23232 . . 3  |-  ( R 
+op  S )  =  ( S  +op  R
)
43oveq1i 6054 . 2  |-  ( ( R  +op  S ) 
+op  T )  =  ( ( S  +op  R )  +op  T )
5 hods.3 . . 3  |-  T : ~H
--> ~H
61, 2, 5hoaddassi 23236 . 2  |-  ( ( R  +op  S ) 
+op  T )  =  ( R  +op  ( S  +op  T ) )
72, 1, 5hoaddassi 23236 . 2  |-  ( ( S  +op  R ) 
+op  T )  =  ( S  +op  ( R  +op  T ) )
84, 6, 73eqtr3i 2436 1  |-  ( R 
+op  ( S  +op  T ) )  =  ( S  +op  ( R 
+op  T ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   -->wf 5413  (class class class)co 6044   ~Hchil 22379    +op chos 22398
This theorem is referenced by:  ho0subi  23255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-hilex 22459  ax-hfvadd 22460  ax-hvcom 22461  ax-hvass 22462
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983  df-hosum 23190
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