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Theorem hvsub4 24590
Description: Hilbert vector space addition/subtraction law. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvsub4  |-  ( ( ( 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 hvsub4
StepHypRef Expression
1 hvaddcl 24565 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
2 hvaddcl 24565 . . 3  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( C  +h  D
)  e.  ~H )
3 hvsubval 24569 . . 3  |-  ( ( ( A  +h  B
)  e.  ~H  /\  ( C  +h  D
)  e.  ~H )  ->  ( ( A  +h  B )  -h  ( C  +h  D ) )  =  ( ( A  +h  B )  +h  ( -u 1  .h  ( C  +h  D
) ) ) )
41, 2, 3syl2an 477 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  -h  ( C  +h  D
) )  =  ( ( A  +h  B
)  +h  ( -u
1  .h  ( C  +h  D ) ) ) )
5 hvsubval 24569 . . . . 5  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  =  ( A  +h  ( -u 1  .h  C ) ) )
65ad2ant2r 746 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( A  -h  C )  =  ( A  +h  ( -u
1  .h  C ) ) )
7 hvsubval 24569 . . . . 5  |-  ( ( B  e.  ~H  /\  D  e.  ~H )  ->  ( B  -h  D
)  =  ( B  +h  ( -u 1  .h  D ) ) )
87ad2ant2l 745 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( B  -h  D )  =  ( B  +h  ( -u
1  .h  D ) ) )
96, 8oveq12d 6217 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  -h  C )  +h  ( B  -h  D
) )  =  ( ( A  +h  ( -u 1  .h  C ) )  +h  ( B  +h  ( -u 1  .h  D ) ) ) )
10 neg1cn 10535 . . . . . . 7  |-  -u 1  e.  CC
11 ax-hvdistr1 24561 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  C  e.  ~H  /\  D  e.  ~H )  ->  ( -u 1  .h  ( C  +h  D
) )  =  ( ( -u 1  .h  C )  +h  ( -u 1  .h  D ) ) )
1210, 11mp3an1 1302 . . . . . 6  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( -u 1  .h  ( C  +h  D
) )  =  ( ( -u 1  .h  C )  +h  ( -u 1  .h  D ) ) )
1312adantl 466 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( -u 1  .h  ( C  +h  D
) )  =  ( ( -u 1  .h  C )  +h  ( -u 1  .h  D ) ) )
1413oveq2d 6215 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  +h  ( -u 1  .h  ( C  +h  D
) ) )  =  ( ( A  +h  B )  +h  (
( -u 1  .h  C
)  +h  ( -u
1  .h  D ) ) ) )
15 hvmulcl 24566 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  C  e.  ~H )  ->  ( -u 1  .h  C )  e.  ~H )
1610, 15mpan 670 . . . . . . . 8  |-  ( C  e.  ~H  ->  ( -u 1  .h  C )  e.  ~H )
1716anim2i 569 . . . . . . 7  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  e.  ~H  /\  ( -u 1  .h  C )  e.  ~H ) )
18 hvmulcl 24566 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  D  e.  ~H )  ->  ( -u 1  .h  D )  e.  ~H )
1910, 18mpan 670 . . . . . . . 8  |-  ( D  e.  ~H  ->  ( -u 1  .h  D )  e.  ~H )
2019anim2i 569 . . . . . . 7  |-  ( ( B  e.  ~H  /\  D  e.  ~H )  ->  ( B  e.  ~H  /\  ( -u 1  .h  D )  e.  ~H ) )
2117, 20anim12i 566 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  C  e.  ~H )  /\  ( B  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  e.  ~H  /\  ( -u 1  .h  C )  e.  ~H )  /\  ( B  e.  ~H  /\  ( -u 1  .h  D )  e.  ~H ) ) )
2221an4s 822 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  e.  ~H  /\  ( -u 1  .h  C )  e.  ~H )  /\  ( B  e.  ~H  /\  ( -u 1  .h  D )  e.  ~H ) ) )
23 hvadd4 24589 . . . . 5  |-  ( ( ( A  e.  ~H  /\  ( -u 1  .h  C )  e.  ~H )  /\  ( B  e. 
~H  /\  ( -u 1  .h  D )  e.  ~H ) )  ->  (
( A  +h  ( -u 1  .h  C ) )  +h  ( B  +h  ( -u 1  .h  D ) ) )  =  ( ( A  +h  B )  +h  ( ( -u 1  .h  C )  +h  ( -u 1  .h  D ) ) ) )
2422, 23syl 16 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  ( -u 1  .h  C ) )  +h  ( B  +h  ( -u 1  .h  D ) ) )  =  ( ( A  +h  B
)  +h  ( (
-u 1  .h  C
)  +h  ( -u
1  .h  D ) ) ) )
2514, 24eqtr4d 2498 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  +h  ( -u 1  .h  ( C  +h  D
) ) )  =  ( ( A  +h  ( -u 1  .h  C
) )  +h  ( B  +h  ( -u 1  .h  D ) ) ) )
269, 25eqtr4d 2498 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  -h  C )  +h  ( B  -h  D
) )  =  ( ( A  +h  B
)  +h  ( -u
1  .h  ( C  +h  D ) ) ) )
274, 26eqtr4d 2498 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6199   CCcc 9390   1c1 9393   -ucneg 9706   ~Hchil 24472    +h cva 24473    .h csm 24474    -h cmv 24478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-hfvadd 24553  ax-hvcom 24554  ax-hvass 24555  ax-hfvmul 24558  ax-hvdistr1 24561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-ltxr 9533  df-sub 9707  df-neg 9708  df-hvsub 24524
This theorem is referenced by:  hvaddsub4  24631  5oalem2  25209  3oalem2  25217
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