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Theorem hvsub4 21929
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 21905 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
2 hvaddcl 21905 . . 3  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( C  +h  D
)  e.  ~H )
3 hvsubval 21909 . . 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 463 . 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 21909 . . . . 5  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  =  ( A  +h  ( -u 1  .h  C ) ) )
65ad2ant2r 727 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( A  -h  C )  =  ( A  +h  ( -u
1  .h  C ) ) )
7 hvsubval 21909 . . . . 5  |-  ( ( B  e.  ~H  /\  D  e.  ~H )  ->  ( B  -h  D
)  =  ( B  +h  ( -u 1  .h  D ) ) )
87ad2ant2l 726 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( B  -h  D )  =  ( B  +h  ( -u
1  .h  D ) ) )
96, 8oveq12d 5999 . . 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 9960 . . . . . . 7  |-  -u 1  e.  CC
11 ax-hvdistr1 21901 . . . . . . 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 1265 . . . . . 6  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( -u 1  .h  ( C  +h  D
) )  =  ( ( -u 1  .h  C )  +h  ( -u 1  .h  D ) ) )
1312adantl 452 . . . . 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 5997 . . . 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 21906 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  C  e.  ~H )  ->  ( -u 1  .h  C )  e.  ~H )
1610, 15mpan 651 . . . . . . . 8  |-  ( C  e.  ~H  ->  ( -u 1  .h  C )  e.  ~H )
1716anim2i 552 . . . . . . 7  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  e.  ~H  /\  ( -u 1  .h  C )  e.  ~H ) )
18 hvmulcl 21906 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  D  e.  ~H )  ->  ( -u 1  .h  D )  e.  ~H )
1910, 18mpan 651 . . . . . . . 8  |-  ( D  e.  ~H  ->  ( -u 1  .h  D )  e.  ~H )
2019anim2i 552 . . . . . . 7  |-  ( ( B  e.  ~H  /\  D  e.  ~H )  ->  ( B  e.  ~H  /\  ( -u 1  .h  D )  e.  ~H ) )
2117, 20anim12i 549 . . . . . 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 799 . . . . 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 21928 . . . . 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 15 . . . 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 2401 . . 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 2401 . 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 2401 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 1647    e. wcel 1715  (class class class)co 5981   CCcc 8882   1c1 8885   -ucneg 9185   ~Hchil 21812    +h cva 21813    .h csm 21814    -h cmv 21818
This theorem is referenced by:  hvaddsub4  21970  5oalem2  22547  3oalem2  22555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-hfvadd 21893  ax-hvcom 21894  ax-hvass 21895  ax-hfvmul 21898  ax-hvdistr1 21901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-ltxr 9019  df-sub 9186  df-neg 9187  df-hvsub 21864
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