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Theorem hvaddsub4 22429
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 22364 . . . 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 22364 . . . 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 22364 . . . . 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 22426 . . 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 22388 . . . . . 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 22377 . . . . . . 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 6037 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A  -h  C )  +h  ( B  -h  B
) )  =  ( ( A  -h  C
)  +h  0h )
)
18 hvsubcl 22369 . . . . . . 7  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  e.  ~H )
19 ax-hvaddid 22356 . . . . . . 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 2424 . . . 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 22388 . . . . . . 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 22377 . . . . . . . 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 6036 . . . . . 6  |-  ( ( ( C  e.  ~H  /\  D  e.  ~H )  /\  B  e.  ~H )  ->  ( ( C  -h  C )  +h  ( D  -h  B
) )  =  ( 0h  +h  ( D  -h  B ) ) )
31 hvsubcl 22369 . . . . . . . 8  |-  ( ( D  e.  ~H  /\  B  e.  ~H )  ->  ( D  -h  B
)  e.  ~H )
32 hvaddid2 22374 . . . . . . . 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 2424 . . . . 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 2402 . 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 1717  (class class class)co 6021   ~Hchil 22271    +h cva 22272   0hc0v 22276    -h cmv 22277
This theorem is referenced by:  shuni  22651  cdjreui  23784
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-hfvadd 22352  ax-hvcom 22353  ax-hvass 22354  ax-hv0cl 22355  ax-hvaddid 22356  ax-hfvmul 22357  ax-hvmulid 22358  ax-hvmulass 22359  ax-hvdistr1 22360  ax-hvdistr2 22361  ax-hvmul0 22362
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-hvsub 22323
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