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Theorem hvaddsub4 22533
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 22468 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
21adantr 452 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( A  +h  B )  e.  ~H )
3 hvaddcl 22468 . . . 4  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  ( C  +h  D
)  e.  ~H )
43adantl 453 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( C  +h  D )  e.  ~H )
5 hvaddcl 22468 . . . . 5  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( C  +h  B
)  e.  ~H )
65ancoms 440 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( C  +h  B
)  e.  ~H )
76ad2ant2lr 729 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( C  +h  B )  e.  ~H )
8 hvsubcan2 22530 . . 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 1184 . 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 448 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  B  e.  ~H )
1110anim2i 553 . . . . . . 7  |-  ( ( C  e.  ~H  /\  ( A  e.  ~H  /\  B  e.  ~H )
)  ->  ( C  e.  ~H  /\  B  e. 
~H ) )
1211ancoms 440 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( C  e. 
~H  /\  B  e.  ~H ) )
13 hvsub4 22492 . . . . . 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 457 . . . . 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 22481 . . . . . . 7  |-  ( B  e.  ~H  ->  ( B  -h  B )  =  0h )
1615ad2antlr 708 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( B  -h  B )  =  0h )
1716oveq2d 6056 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  -h  C )  +h  ( B  -h  B
) )  =  ( ( A  -h  C
)  +h  0h )
)
18 hvsubcl 22473 . . . . . . 7  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  e.  ~H )
19 ax-hvaddid 22460 . . . . . . 7  |-  ( ( A  -h  C )  e.  ~H  ->  (
( A  -h  C
)  +h  0h )  =  ( A  -h  C ) )
2018, 19syl 16 . . . . . 6  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  C )  +h  0h )  =  ( A  -h  C ) )
2120adantlr 696 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  -h  C )  +h 
0h )  =  ( A  -h  C ) )
2214, 17, 213eqtrd 2440 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  +h  B )  -h  ( C  +h  B
) )  =  ( A  -h  C ) )
2322adantrr 698 . . 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 444 . . . . . . . 8  |-  ( ( C  e.  ~H  /\  D  e.  ~H )  ->  C  e.  ~H )
2524anim1i 552 . . . . . . 7  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( C  e. 
~H  /\  B  e.  ~H ) )
26 hvsub4 22492 . . . . . . 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 457 . . . . . 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 22481 . . . . . . . 8  |-  ( C  e.  ~H  ->  ( C  -h  C )  =  0h )
2928ad2antrr 707 . . . . . . 7  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( C  -h  C )  =  0h )
3029oveq1d 6055 . . . . . 6  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( ( C  -h  C )  +h  ( D  -h  B
) )  =  ( 0h  +h  ( D  -h  B ) ) )
31 hvsubcl 22473 . . . . . . . 8  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( D  -h  B
)  e.  ~H )
32 hvaddid2 22478 . . . . . . . 8  |-  ( ( D  -h  B )  e.  ~H  ->  ( 0h  +h  ( D  -h  B ) )  =  ( D  -h  B
) )
3331, 32syl 16 . . . . . . 7  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( 0h  +h  ( D  -h  B ) )  =  ( D  -h  B ) )
3433adantll 695 . . . . . 6  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( 0h  +h  ( D  -h  B
) )  =  ( D  -h  B ) )
3527, 30, 343eqtrd 2440 . . . . 5  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( ( C  +h  D )  -h  ( C  +h  B
) )  =  ( D  -h  B ) )
3635ancoms 440 . . . 4  |-  ( ( B  e.  ~H  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( C  +h  D )  -h  ( C  +h  B
) )  =  ( D  -h  B ) )
3736adantll 695 . . 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 2418 . 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 247 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 177    /\ wa 359    = wceq 1649    e. wcel 1721  (class class class)co 6040   ~Hchil 22375    +h cva 22376   0hc0v 22380    -h cmv 22381
This theorem is referenced by:  shuni  22755  cdjreui  23888
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-pre-mulgt0 9023  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466
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-rmo 2674  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-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-hvsub 22427
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