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Theorem hvsub4 21618
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 21594 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
2 hvaddcl 21594 . . 3  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( C  +h  D
)  e.  ~H )
3 hvsubval 21598 . . 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 21598 . . . . 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 21598 . . . . 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 5878 . . 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 9815 . . . . . . 7  |-  -u 1  e.  CC
11 ax-hvdistr1 21590 . . . . . . 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 1264 . . . . . 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 5876 . . . 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 21595 . . . . . . . . 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 21595 . . . . . . . . 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 21617 . . . . 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 2320 . . 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 2320 . 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 2320 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 1625    e. wcel 1686  (class class class)co 5860   CCcc 8737   1c1 8740   -ucneg 9040   ~Hchil 21501    +h cva 21502    .h csm 21503    -h cmv 21507
This theorem is referenced by:  hvaddsub4  21659  5oalem2  22236  3oalem2  22244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-hfvadd 21582  ax-hvcom 21583  ax-hvass 21584  ax-hfvmul 21587  ax-hvdistr1 21590
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-ltxr 8874  df-sub 9041  df-neg 9042  df-hvsub 21553
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