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Theorem hvsub32 24382
Description: Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hvsub32  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  -h  C )  =  ( ( A  -h  C )  -h  B ) )

Proof of Theorem hvsub32
StepHypRef Expression
1 ax-hvcom 24338 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  =  ( C  +h  B ) )
213adant1 1001 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C )  =  ( C  +h  B
) )
32oveq2d 6106 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  ( B  +h  C ) )  =  ( A  -h  ( C  +h  B ) ) )
4 hvsubass 24381 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  -h  C )  =  ( A  -h  ( B  +h  C
) ) )
5 hvsubass 24381 . . 3  |-  ( ( A  e.  ~H  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
( A  -h  C
)  -h  B )  =  ( A  -h  ( C  +h  B
) ) )
653com23 1188 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  C
)  -h  B )  =  ( A  -h  ( C  +h  B
) ) )
73, 4, 63eqtr4d 2483 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  -h  C )  =  ( ( A  -h  C )  -h  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 960    = wceq 1364    e. wcel 1761  (class class class)co 6090   ~Hchil 24256    +h cva 24257    -h cmv 24262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-hfvadd 24337  ax-hvcom 24338  ax-hvass 24339  ax-hfvmul 24342  ax-hvdistr1 24345
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-neg 9594  df-hvsub 24308
This theorem is referenced by:  hvsub32i  24393
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