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Theorem hoadd12i 22465
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 22460 . . 3  |-  ( R 
+op  S )  =  ( S  +op  R
)
43oveq1i 5952 . 2  |-  ( ( R  +op  S ) 
+op  T )  =  ( ( S  +op  R )  +op  T )
5 hods.3 . . 3  |-  T : ~H
--> ~H
61, 2, 5hoaddassi 22464 . 2  |-  ( ( R  +op  S ) 
+op  T )  =  ( R  +op  ( S  +op  T ) )
72, 1, 5hoaddassi 22464 . 2  |-  ( ( S  +op  R ) 
+op  T )  =  ( S  +op  ( R  +op  T ) )
84, 6, 73eqtr3i 2386 1  |-  ( R 
+op  ( S  +op  T ) )  =  ( S  +op  ( R 
+op  T ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1642   -->wf 5330  (class class class)co 5942   ~Hchil 21607    +op chos 21626
This theorem is referenced by:  ho0subi  22483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-hilex 21687  ax-hfvadd 21688  ax-hvcom 21689  ax-hvass 21690
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-map 6859  df-hosum 22418
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