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Theorem hvsub4 26096
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 26071 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
2 hvaddcl 26071 . . 3  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( C  +h  D
)  e.  ~H )
3 hvsubval 26075 . . 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 475 . 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 26075 . . . . 5  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  =  ( A  +h  ( -u 1  .h  C ) ) )
65ad2ant2r 744 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( A  -h  C )  =  ( A  +h  ( -u
1  .h  C ) ) )
7 hvsubval 26075 . . . . 5  |-  ( ( B  e.  ~H  /\  D  e.  ~H )  ->  ( B  -h  D
)  =  ( B  +h  ( -u 1  .h  D ) ) )
87ad2ant2l 743 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( B  -h  D )  =  ( B  +h  ( -u
1  .h  D ) ) )
96, 8oveq12d 6236 . . 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 10578 . . . . . . 7  |-  -u 1  e.  CC
11 ax-hvdistr1 26067 . . . . . . 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 1309 . . . . . 6  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( -u 1  .h  ( C  +h  D
) )  =  ( ( -u 1  .h  C )  +h  ( -u 1  .h  D ) ) )
1312adantl 464 . . . . 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 6234 . . . 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 26072 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  C  e.  ~H )  ->  ( -u 1  .h  C )  e.  ~H )
1610, 15mpan 668 . . . . . . . 8  |-  ( C  e.  ~H  ->  ( -u 1  .h  C )  e.  ~H )
1716anim2i 567 . . . . . . 7  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  e.  ~H  /\  ( -u 1  .h  C )  e.  ~H ) )
18 hvmulcl 26072 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  D  e.  ~H )  ->  ( -u 1  .h  D )  e.  ~H )
1910, 18mpan 668 . . . . . . . 8  |-  ( D  e.  ~H  ->  ( -u 1  .h  D )  e.  ~H )
2019anim2i 567 . . . . . . 7  |-  ( ( B  e.  ~H  /\  D  e.  ~H )  ->  ( B  e.  ~H  /\  ( -u 1  .h  D )  e.  ~H ) )
2117, 20anim12i 564 . . . . . 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 824 . . . . 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 26095 . . . . 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 2440 . . 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 2440 . 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 2440 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 367    = wceq 1399    e. wcel 1836  (class class class)co 6218   CCcc 9423   1c1 9426   -ucneg 9741   ~Hchil 25978    +h cva 25979    .h csm 25980    -h cmv 25984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-hfvadd 26059  ax-hvcom 26060  ax-hvass 26061  ax-hfvmul 26064  ax-hvdistr1 26067
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-po 4731  df-so 4732  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-ltxr 9566  df-sub 9742  df-neg 9743  df-hvsub 26030
This theorem is referenced by:  hvaddsub4  26137  5oalem2  26715  3oalem2  26723
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