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Theorem hvsubdistr2 22400
Description: Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubdistr2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  -  B
)  .h  C )  =  ( ( A  .h  C )  -h  ( B  .h  C
) ) )

Proof of Theorem hvsubdistr2
StepHypRef Expression
1 hvmulcl 22364 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
213adant2 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C )  e.  ~H )
3 hvmulcl 22364 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  ~H )  ->  ( B  .h  C
)  e.  ~H )
433adant1 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( B  .h  C )  e.  ~H )
5 hvsubval 22367 . . 3  |-  ( ( ( A  .h  C
)  e.  ~H  /\  ( B  .h  C
)  e.  ~H )  ->  ( ( A  .h  C )  -h  ( B  .h  C )
)  =  ( ( A  .h  C )  +h  ( -u 1  .h  ( B  .h  C
) ) ) )
62, 4, 5syl2anc 643 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  .h  C
)  -h  ( B  .h  C ) )  =  ( ( A  .h  C )  +h  ( -u 1  .h  ( B  .h  C
) ) ) )
7 mulm1 9407 . . . . . . 7  |-  ( B  e.  CC  ->  ( -u 1  x.  B )  =  -u B )
87oveq1d 6035 . . . . . 6  |-  ( B  e.  CC  ->  (
( -u 1  x.  B
)  .h  C )  =  ( -u B  .h  C ) )
98adantr 452 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  ~H )  ->  ( ( -u 1  x.  B )  .h  C
)  =  ( -u B  .h  C )
)
10 neg1cn 9999 . . . . . 6  |-  -u 1  e.  CC
11 ax-hvmulass 22358 . . . . . 6  |-  ( (
-u 1  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( -u 1  x.  B )  .h  C
)  =  ( -u
1  .h  ( B  .h  C ) ) )
1210, 11mp3an1 1266 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  ~H )  ->  ( ( -u 1  x.  B )  .h  C
)  =  ( -u
1  .h  ( B  .h  C ) ) )
139, 12eqtr3d 2421 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  ~H )  ->  ( -u B  .h  C )  =  (
-u 1  .h  ( B  .h  C )
) )
14133adant1 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( -u B  .h  C )  =  ( -u 1  .h  ( B  .h  C
) ) )
1514oveq2d 6036 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  .h  C
)  +h  ( -u B  .h  C )
)  =  ( ( A  .h  C )  +h  ( -u 1  .h  ( B  .h  C
) ) ) )
16 negcl 9238 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
17 ax-hvdistr2 22360 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  +  -u B )  .h  C
)  =  ( ( A  .h  C )  +h  ( -u B  .h  C ) ) )
1816, 17syl3an2 1218 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  -u B )  .h  C
)  =  ( ( A  .h  C )  +h  ( -u B  .h  C ) ) )
19 negsub 9281 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
20193adant3 977 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  +  -u B )  =  ( A  -  B ) )
2120oveq1d 6035 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  -u B )  .h  C
)  =  ( ( A  -  B )  .h  C ) )
2218, 21eqtr3d 2421 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  .h  C
)  +h  ( -u B  .h  C )
)  =  ( ( A  -  B )  .h  C ) )
236, 15, 223eqtr2rd 2426 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  -  B
)  .h  C )  =  ( ( A  .h  C )  -h  ( B  .h  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6020   CCcc 8921   1c1 8924    + caddc 8926    x. cmul 8928    - cmin 9223   -ucneg 9224   ~Hchil 22270    +h cva 22271    .h csm 22272    -h cmv 22276
This theorem is referenced by:  hvmulcan2  22423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-hfvmul 22356  ax-hvmulass 22358  ax-hvdistr2 22360
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-neg 9226  df-hvsub 22322
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