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Theorem hvsubsub4 22562
Description: Hilbert vector space addition/subtraction law. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubsub4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  -h  B )  -h  ( C  -h  D
) )  =  ( ( A  -h  C
)  -h  ( B  -h  D ) ) )

Proof of Theorem hvsubsub4
StepHypRef Expression
1 oveq1 6088 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21oveq1d 6096 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  -h  B
)  -h  ( C  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  -h  ( C  -h  D ) ) )
3 oveq1 6088 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )
43oveq1d 6096 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  -h  C
)  -h  ( B  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  -h  ( B  -h  D ) ) )
52, 4eqeq12d 2450 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( A  -h  B )  -h  ( C  -h  D ) )  =  ( ( A  -h  C )  -h  ( B  -h  D
) )  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  -h  ( C  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  -h  ( B  -h  D ) ) ) )
6 oveq2 6089 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )
76oveq1d 6096 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  B
)  -h  ( C  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  -h  ( C  -h  D ) ) )
8 oveq1 6088 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  -h  D )  =  ( if ( B  e.  ~H ,  B ,  0h )  -h  D
) )
98oveq2d 6097 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  C
)  -h  ( B  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  D ) ) )
107, 9eqeq12d 2450 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  -h  ( C  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  -h  ( B  -h  D ) )  <-> 
( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  -h  ( C  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  D ) ) ) )
11 oveq1 6088 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( C  -h  D )  =  ( if ( C  e.  ~H ,  C ,  0h )  -h  D
) )
1211oveq2d 6097 . . 3  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
)  -h  ( C  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  -h  ( if ( C  e.  ~H ,  C ,  0h )  -h  D ) ) )
13 oveq2 6089 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( C  e.  ~H ,  C ,  0h )
) )
1413oveq1d 6096 . . 3  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  C
)  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  D ) ) )
1512, 14eqeq12d 2450 . 2  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  -h  ( C  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  C )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  D ) )  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  -h  ( if ( C  e.  ~H ,  C ,  0h )  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  D ) ) ) )
16 oveq2 6089 . . . 4  |-  ( D  =  if ( D  e.  ~H ,  D ,  0h )  ->  ( if ( C  e.  ~H ,  C ,  0h )  -h  D )  =  ( if ( C  e. 
~H ,  C ,  0h )  -h  if ( D  e.  ~H ,  D ,  0h )
) )
1716oveq2d 6097 . . 3  |-  ( D  =  if ( D  e.  ~H ,  D ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
)  -h  ( if ( C  e.  ~H ,  C ,  0h )  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  -h  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( D  e. 
~H ,  D ,  0h ) ) ) )
18 oveq2 6089 . . . 4  |-  ( D  =  if ( D  e.  ~H ,  D ,  0h )  ->  ( if ( B  e.  ~H ,  B ,  0h )  -h  D )  =  ( if ( B  e. 
~H ,  B ,  0h )  -h  if ( D  e.  ~H ,  D ,  0h )
) )
1918oveq2d 6097 . . 3  |-  ( D  =  if ( D  e.  ~H ,  D ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e.  ~H ,  C ,  0h )
)  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  if ( D  e. 
~H ,  D ,  0h ) ) ) )
2017, 19eqeq12d 2450 . 2  |-  ( D  =  if ( D  e.  ~H ,  D ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  -h  ( if ( C  e.  ~H ,  C ,  0h )  -h  D ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  D ) )  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  -h  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( D  e. 
~H ,  D ,  0h ) ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  if ( D  e. 
~H ,  D ,  0h ) ) ) ) )
21 ax-hv0cl 22506 . . . 4  |-  0h  e.  ~H
2221elimel 3791 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
2321elimel 3791 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
2421elimel 3791 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
2521elimel 3791 . . 3  |-  if ( D  e.  ~H ,  D ,  0h )  e.  ~H
2622, 23, 24, 25hvsubsub4i 22561 . 2  |-  ( ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
)  -h  ( if ( C  e.  ~H ,  C ,  0h )  -h  if ( D  e. 
~H ,  D ,  0h ) ) )  =  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( C  e. 
~H ,  C ,  0h ) )  -h  ( if ( B  e.  ~H ,  B ,  0h )  -h  if ( D  e. 
~H ,  D ,  0h ) ) )
275, 10, 15, 20, 26dedth4h 3783 1  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  -h  B )  -h  ( C  -h  D
) )  =  ( ( A  -h  C
)  -h  ( B  -h  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3739  (class class class)co 6081   ~Hchil 22422   0hc0v 22427    -h cmv 22428
This theorem is referenced by:  chscllem2  23140  5oalem3  23158  5oalem5  23160
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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hfvmul 22508  ax-hvdistr1 22511
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2602  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-po 4503  df-so 4504  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-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-hvsub 22474
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