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Theorem hoadd32i 23281
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 23275 . . 3  |-  ( S 
+op  T )  =  ( T  +op  S
)
43oveq2i 6092 . 2  |-  ( R 
+op  ( S  +op  T ) )  =  ( R  +op  ( T 
+op  S ) )
5 hods.1 . . 3  |-  R : ~H
--> ~H
65, 1, 2hoaddassi 23279 . 2  |-  ( ( R  +op  S ) 
+op  T )  =  ( R  +op  ( S  +op  T ) )
75, 2, 1hoaddassi 23279 . 2  |-  ( ( R  +op  T ) 
+op  S )  =  ( R  +op  ( T  +op  S ) )
84, 6, 73eqtr4i 2466 1  |-  ( ( R  +op  S ) 
+op  T )  =  ( ( R  +op  T )  +op  S )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   -->wf 5450  (class class class)co 6081   ~Hchil 22422    +op chos 22441
This theorem is referenced by:  hosubeq0i  23329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-hilex 22502  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-hosum 23233
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