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Theorem hvsubsub4 21639
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 5865 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21oveq1d 5873 . . 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 5865 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  C
) )
43oveq1d 5873 . . 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 2297 . 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 5866 . . . 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 5873 . . 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 5865 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  -h  D )  =  ( if ( B  e.  ~H ,  B ,  0h )  -h  D
) )
98oveq2d 5874 . . 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 2297 . 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 5865 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( C  -h  D )  =  ( if ( C  e.  ~H ,  C ,  0h )  -h  D
) )
1211oveq2d 5874 . . 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 5866 . . . 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 5873 . . 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 2297 . 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 5866 . . . 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 5874 . . 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 5866 . . . 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 5874 . . 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 2297 . 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 21583 . . . 4  |-  0h  e.  ~H
2221elimel 3617 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
2321elimel 3617 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
2421elimel 3617 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
2521elimel 3617 . . 3  |-  if ( D  e.  ~H ,  D ,  0h )  e.  ~H
2622, 23, 24, 25hvsubsub4i 21638 . 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 3609 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 358    = wceq 1623    e. wcel 1684   ifcif 3565  (class class class)co 5858   ~Hchil 21499   0hc0v 21504    -h cmv 21505
This theorem is referenced by:  chscllem2  22217  5oalem3  22235  5oalem5  22237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hfvmul 21585  ax-hvdistr1 21588
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-hvsub 21551
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