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Theorem hvnegdii 22564
Description: Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvnegdi.1  |-  A  e. 
~H
hvnegdi.2  |-  B  e. 
~H
Assertion
Ref Expression
hvnegdii  |-  ( -u
1  .h  ( A  -h  B ) )  =  ( B  -h  A )

Proof of Theorem hvnegdii
StepHypRef Expression
1 hvnegdi.1 . . . 4  |-  A  e. 
~H
2 hvnegdi.2 . . . 4  |-  B  e. 
~H
31, 2hvsubvali 22523 . . 3  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
43oveq2i 6092 . 2  |-  ( -u
1  .h  ( A  -h  B ) )  =  ( -u 1  .h  ( A  +h  ( -u 1  .h  B ) ) )
5 neg1cn 10067 . . 3  |-  -u 1  e.  CC
65, 2hvmulcli 22517 . . 3  |-  ( -u
1  .h  B )  e.  ~H
75, 1, 6hvdistr1i 22553 . 2  |-  ( -u
1  .h  ( A  +h  ( -u 1  .h  B ) ) )  =  ( ( -u
1  .h  A )  +h  ( -u 1  .h  ( -u 1  .h  B ) ) )
8 ax-1cn 9048 . . . . . . . 8  |-  1  e.  CC
98, 8mul2negi 9481 . . . . . . 7  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
10 1t1e1 10126 . . . . . . 7  |-  ( 1  x.  1 )  =  1
119, 10eqtri 2456 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
1211oveq1i 6091 . . . . 5  |-  ( (
-u 1  x.  -u 1
)  .h  B )  =  ( 1  .h  B )
135, 5, 2hvmulassi 22548 . . . . 5  |-  ( (
-u 1  x.  -u 1
)  .h  B )  =  ( -u 1  .h  ( -u 1  .h  B ) )
14 ax-hvmulid 22509 . . . . . 6  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
152, 14ax-mp 8 . . . . 5  |-  ( 1  .h  B )  =  B
1612, 13, 153eqtr3i 2464 . . . 4  |-  ( -u
1  .h  ( -u
1  .h  B ) )  =  B
1716oveq1i 6091 . . 3  |-  ( (
-u 1  .h  ( -u 1  .h  B ) )  +h  ( -u
1  .h  A ) )  =  ( B  +h  ( -u 1  .h  A ) )
185, 1hvmulcli 22517 . . . 4  |-  ( -u
1  .h  A )  e.  ~H
195, 6hvmulcli 22517 . . . 4  |-  ( -u
1  .h  ( -u
1  .h  B ) )  e.  ~H
2018, 19hvcomi 22522 . . 3  |-  ( (
-u 1  .h  A
)  +h  ( -u
1  .h  ( -u
1  .h  B ) ) )  =  ( ( -u 1  .h  ( -u 1  .h  B ) )  +h  ( -u 1  .h  A ) )
212, 1hvsubvali 22523 . . 3  |-  ( B  -h  A )  =  ( B  +h  ( -u 1  .h  A ) )
2217, 20, 213eqtr4i 2466 . 2  |-  ( (
-u 1  .h  A
)  +h  ( -u
1  .h  ( -u
1  .h  B ) ) )  =  ( B  -h  A )
234, 7, 223eqtri 2460 1  |-  ( -u
1  .h  ( A  -h  B ) )  =  ( B  -h  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725  (class class class)co 6081   1c1 8991    x. cmul 8995   -ucneg 9292   ~Hchil 22422    +h cva 22423    .h csm 22424    -h cmv 22428
This theorem is referenced by:  hvnegdi  22569  hisubcomi  22606  normsubi  22643  normpar2i  22658  pjsslem  23181  pjcji  23186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-hvcom 22504  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmulass 22510  ax-hvdistr1 22511
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-hvsub 22474
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