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Theorem hvsubdistr2 22505
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 22469 . . . 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 22469 . . . 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 22472 . . 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 9431 . . . . . . 7  |-  ( B  e.  CC  ->  ( -u 1  x.  B )  =  -u B )
87oveq1d 6055 . . . . . 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 10023 . . . . . 6  |-  -u 1  e.  CC
11 ax-hvmulass 22463 . . . . . 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 2438 . . . 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 6056 . 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 9262 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
17 ax-hvdistr2 22465 . . . 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 9305 . . . . 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 6055 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  -u B )  .h  C
)  =  ( ( A  -  B )  .h  C ) )
2218, 21eqtr3d 2438 . 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 2443 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 1721  (class class class)co 6040   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248   ~Hchil 22375    +h cva 22376    .h csm 22377    -h cmv 22381
This theorem is referenced by:  hvmulcan2  22528
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hfvmul 22461  ax-hvmulass 22463  ax-hvdistr2 22465
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-hvsub 22427
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