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Theorem hvaddsub4 21673
Description: Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvaddsub4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  =  ( C  +h  D
)  <->  ( A  -h  C )  =  ( D  -h  B ) ) )

Proof of Theorem hvaddsub4
StepHypRef Expression
1 hvaddcl 21608 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
21adantr 451 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( A  +h  B )  e.  ~H )
3 hvaddcl 21608 . . . 4  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( C  +h  D
)  e.  ~H )
43adantl 452 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( C  +h  D )  e.  ~H )
5 hvaddcl 21608 . . . . 5  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( C  +h  B
)  e.  ~H )
65ancoms 439 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( C  +h  B
)  e.  ~H )
76ad2ant2lr 728 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( C  +h  B )  e.  ~H )
8 hvsubcan2 21670 . . 3  |-  ( ( ( A  +h  B
)  e.  ~H  /\  ( C  +h  D
)  e.  ~H  /\  ( C  +h  B
)  e.  ~H )  ->  ( ( ( A  +h  B )  -h  ( C  +h  B
) )  =  ( ( C  +h  D
)  -h  ( C  +h  B ) )  <-> 
( A  +h  B
)  =  ( C  +h  D ) ) )
92, 4, 7, 8syl3anc 1182 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  +h  B
)  -h  ( C  +h  B ) )  =  ( ( C  +h  D )  -h  ( C  +h  B
) )  <->  ( A  +h  B )  =  ( C  +h  D ) ) )
10 simpr 447 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  B  e.  ~H )
1110anim2i 552 . . . . . . 7  |-  ( ( C  e.  ~H  /\  ( A  e.  ~H  /\  B  e.  ~H )
)  ->  ( C  e.  ~H  /\  B  e. 
~H ) )
1211ancoms 439 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( C  e. 
~H  /\  B  e.  ~H ) )
13 hvsub4 21632 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  B  e.  ~H )
)  ->  ( ( A  +h  B )  -h  ( C  +h  B
) )  =  ( ( A  -h  C
)  +h  ( B  -h  B ) ) )
1412, 13syldan 456 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  +h  B )  -h  ( C  +h  B
) )  =  ( ( A  -h  C
)  +h  ( B  -h  B ) ) )
15 hvsubid 21621 . . . . . . 7  |-  ( B  e.  ~H  ->  ( B  -h  B )  =  0h )
1615ad2antlr 707 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( B  -h  B )  =  0h )
1716oveq2d 5890 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  -h  C )  +h  ( B  -h  B
) )  =  ( ( A  -h  C
)  +h  0h )
)
18 hvsubcl 21613 . . . . . . 7  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  e.  ~H )
19 ax-hvaddid 21600 . . . . . . 7  |-  ( ( A  -h  C )  e.  ~H  ->  (
( A  -h  C
)  +h  0h )  =  ( A  -h  C ) )
2018, 19syl 15 . . . . . 6  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  C )  +h  0h )  =  ( A  -h  C ) )
2120adantlr 695 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  -h  C )  +h 
0h )  =  ( A  -h  C ) )
2214, 17, 213eqtrd 2332 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  +h  B )  -h  ( C  +h  B
) )  =  ( A  -h  C ) )
2322adantrr 697 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  -h  ( C  +h  B
) )  =  ( A  -h  C ) )
24 simpl 443 . . . . . . . 8  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  C  e.  ~H )
2524anim1i 551 . . . . . . 7  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( C  e. 
~H  /\  B  e.  ~H ) )
26 hvsub4 21632 . . . . . . 7  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  ( C  e.  ~H  /\  B  e.  ~H )
)  ->  ( ( C  +h  D )  -h  ( C  +h  B
) )  =  ( ( C  -h  C
)  +h  ( D  -h  B ) ) )
2725, 26syldan 456 . . . . . 6  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( ( C  +h  D )  -h  ( C  +h  B
) )  =  ( ( C  -h  C
)  +h  ( D  -h  B ) ) )
28 hvsubid 21621 . . . . . . . 8  |-  ( C  e.  ~H  ->  ( C  -h  C )  =  0h )
2928ad2antrr 706 . . . . . . 7  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( C  -h  C )  =  0h )
3029oveq1d 5889 . . . . . 6  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( ( C  -h  C )  +h  ( D  -h  B
) )  =  ( 0h  +h  ( D  -h  B ) ) )
31 hvsubcl 21613 . . . . . . . 8  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( D  -h  B
)  e.  ~H )
32 hvaddid2 21618 . . . . . . . 8  |-  ( ( D  -h  B )  e.  ~H  ->  ( 0h  +h  ( D  -h  B ) )  =  ( D  -h  B
) )
3331, 32syl 15 . . . . . . 7  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( 0h  +h  ( D  -h  B ) )  =  ( D  -h  B ) )
3433adantll 694 . . . . . 6  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( 0h  +h  ( D  -h  B
) )  =  ( D  -h  B ) )
3527, 30, 343eqtrd 2332 . . . . 5  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( ( C  +h  D )  -h  ( C  +h  B
) )  =  ( D  -h  B ) )
3635ancoms 439 . . . 4  |-  ( ( B  e.  ~H  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( C  +h  D )  -h  ( C  +h  B
) )  =  ( D  -h  B ) )
3736adantll 694 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( C  +h  D )  -h  ( C  +h  B
) )  =  ( D  -h  B ) )
3823, 37eqeq12d 2310 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  +h  B
)  -h  ( C  +h  B ) )  =  ( ( C  +h  D )  -h  ( C  +h  B
) )  <->  ( A  -h  C )  =  ( D  -h  B ) ) )
399, 38bitr3d 246 1  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  =  ( C  +h  D
)  <->  ( A  -h  C )  =  ( D  -h  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696  (class class class)co 5874   ~Hchil 21515    +h cva 21516   0hc0v 21520    -h cmv 21521
This theorem is referenced by:  shuni  21895  cdjreui  23028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-hvsub 21567
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