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Theorem hvadd12i 22561
Description: Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvass.1  |-  A  e. 
~H
hvass.2  |-  B  e. 
~H
hvass.3  |-  C  e. 
~H
Assertion
Ref Expression
hvadd12i  |-  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) )

Proof of Theorem hvadd12i
StepHypRef Expression
1 hvass.1 . . . 4  |-  A  e. 
~H
2 hvass.2 . . . 4  |-  B  e. 
~H
31, 2hvcomi 22524 . . 3  |-  ( A  +h  B )  =  ( B  +h  A
)
43oveq1i 6093 . 2  |-  ( ( A  +h  B )  +h  C )  =  ( ( B  +h  A )  +h  C
)
5 hvass.3 . . 3  |-  C  e. 
~H
61, 2, 5hvassi 22557 . 2  |-  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) )
72, 1, 5hvassi 22557 . 2  |-  ( ( B  +h  A )  +h  C )  =  ( B  +h  ( A  +h  C ) )
84, 6, 73eqtr3i 2466 1  |-  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726  (class class class)co 6083   ~Hchil 22424    +h cva 22425
This theorem is referenced by:  hvsubaddi  22570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-hvcom 22506  ax-hvass 22507
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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