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Theorem hvaddsub4 21657
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 21592 . . . 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 21592 . . . 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 21592 . . . . 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 21654 . . 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 21616 . . . . . 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 21605 . . . . . . 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 5874 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  -h  C )  +h  ( B  -h  B
) )  =  ( ( A  -h  C
)  +h  0h )
)
18 hvsubcl 21597 . . . . . . 7  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  e.  ~H )
19 ax-hvaddid 21584 . . . . . . 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 2319 . . . 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 21616 . . . . . . 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 21605 . . . . . . . 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 5873 . . . . . 6  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( ( C  -h  C )  +h  ( D  -h  B
) )  =  ( 0h  +h  ( D  -h  B ) ) )
31 hvsubcl 21597 . . . . . . . 8  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( D  -h  B
)  e.  ~H )
32 hvaddid2 21602 . . . . . . . 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 2319 . . . . 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 2297 . 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 1623    e. wcel 1684  (class class class)co 5858   ~Hchil 21499    +h cva 21500   0hc0v 21504    -h cmv 21505
This theorem is referenced by:  shuni  21879  cdjreui  23012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-hvsub 21551
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