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Theorem hvsubsub4i 22514
Description: Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvass.1  |-  A  e. 
~H
hvass.2  |-  B  e. 
~H
hvass.3  |-  C  e. 
~H
hvadd4.4  |-  D  e. 
~H
Assertion
Ref Expression
hvsubsub4i  |-  ( ( A  -h  B )  -h  ( C  -h  D ) )  =  ( ( A  -h  C )  -h  ( B  -h  D ) )

Proof of Theorem hvsubsub4i
StepHypRef Expression
1 hvass.1 . . . . 5  |-  A  e. 
~H
2 neg1cn 10023 . . . . . 6  |-  -u 1  e.  CC
3 hvass.2 . . . . . 6  |-  B  e. 
~H
42, 3hvmulcli 22470 . . . . 5  |-  ( -u
1  .h  B )  e.  ~H
5 hvass.3 . . . . . 6  |-  C  e. 
~H
62, 5hvmulcli 22470 . . . . 5  |-  ( -u
1  .h  C )  e.  ~H
7 hvadd4.4 . . . . . . 7  |-  D  e. 
~H
82, 7hvmulcli 22470 . . . . . 6  |-  ( -u
1  .h  D )  e.  ~H
92, 8hvmulcli 22470 . . . . 5  |-  ( -u
1  .h  ( -u
1  .h  D ) )  e.  ~H
101, 4, 6, 9hvadd4i 22513 . . . 4  |-  ( ( A  +h  ( -u
1  .h  B ) )  +h  ( (
-u 1  .h  C
)  +h  ( -u
1  .h  ( -u
1  .h  D ) ) ) )  =  ( ( A  +h  ( -u 1  .h  C
) )  +h  (
( -u 1  .h  B
)  +h  ( -u
1  .h  ( -u
1  .h  D ) ) ) )
112, 5, 8hvdistr1i 22506 . . . . 5  |-  ( -u
1  .h  ( C  +h  ( -u 1  .h  D ) ) )  =  ( ( -u
1  .h  C )  +h  ( -u 1  .h  ( -u 1  .h  D ) ) )
1211oveq2i 6051 . . . 4  |-  ( ( A  +h  ( -u
1  .h  B ) )  +h  ( -u
1  .h  ( C  +h  ( -u 1  .h  D ) ) ) )  =  ( ( A  +h  ( -u
1  .h  B ) )  +h  ( (
-u 1  .h  C
)  +h  ( -u
1  .h  ( -u
1  .h  D ) ) ) )
132, 3, 8hvdistr1i 22506 . . . . 5  |-  ( -u
1  .h  ( B  +h  ( -u 1  .h  D ) ) )  =  ( ( -u
1  .h  B )  +h  ( -u 1  .h  ( -u 1  .h  D ) ) )
1413oveq2i 6051 . . . 4  |-  ( ( A  +h  ( -u
1  .h  C ) )  +h  ( -u
1  .h  ( B  +h  ( -u 1  .h  D ) ) ) )  =  ( ( A  +h  ( -u
1  .h  C ) )  +h  ( (
-u 1  .h  B
)  +h  ( -u
1  .h  ( -u
1  .h  D ) ) ) )
1510, 12, 143eqtr4i 2434 . . 3  |-  ( ( A  +h  ( -u
1  .h  B ) )  +h  ( -u
1  .h  ( C  +h  ( -u 1  .h  D ) ) ) )  =  ( ( A  +h  ( -u
1  .h  C ) )  +h  ( -u
1  .h  ( B  +h  ( -u 1  .h  D ) ) ) )
161, 4hvaddcli 22474 . . . 4  |-  ( A  +h  ( -u 1  .h  B ) )  e. 
~H
175, 8hvaddcli 22474 . . . 4  |-  ( C  +h  ( -u 1  .h  D ) )  e. 
~H
1816, 17hvsubvali 22476 . . 3  |-  ( ( A  +h  ( -u
1  .h  B ) )  -h  ( C  +h  ( -u 1  .h  D ) ) )  =  ( ( A  +h  ( -u 1  .h  B ) )  +h  ( -u 1  .h  ( C  +h  ( -u 1  .h  D ) ) ) )
191, 6hvaddcli 22474 . . . 4  |-  ( A  +h  ( -u 1  .h  C ) )  e. 
~H
203, 8hvaddcli 22474 . . . 4  |-  ( B  +h  ( -u 1  .h  D ) )  e. 
~H
2119, 20hvsubvali 22476 . . 3  |-  ( ( A  +h  ( -u
1  .h  C ) )  -h  ( B  +h  ( -u 1  .h  D ) ) )  =  ( ( A  +h  ( -u 1  .h  C ) )  +h  ( -u 1  .h  ( B  +h  ( -u 1  .h  D ) ) ) )
2215, 18, 213eqtr4i 2434 . 2  |-  ( ( A  +h  ( -u
1  .h  B ) )  -h  ( C  +h  ( -u 1  .h  D ) ) )  =  ( ( A  +h  ( -u 1  .h  C ) )  -h  ( B  +h  ( -u 1  .h  D ) ) )
231, 3hvsubvali 22476 . . 3  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
245, 7hvsubvali 22476 . . 3  |-  ( C  -h  D )  =  ( C  +h  ( -u 1  .h  D ) )
2523, 24oveq12i 6052 . 2  |-  ( ( A  -h  B )  -h  ( C  -h  D ) )  =  ( ( A  +h  ( -u 1  .h  B
) )  -h  ( C  +h  ( -u 1  .h  D ) ) )
261, 5hvsubvali 22476 . . 3  |-  ( A  -h  C )  =  ( A  +h  ( -u 1  .h  C ) )
273, 7hvsubvali 22476 . . 3  |-  ( B  -h  D )  =  ( B  +h  ( -u 1  .h  D ) )
2826, 27oveq12i 6052 . 2  |-  ( ( A  -h  C )  -h  ( B  -h  D ) )  =  ( ( A  +h  ( -u 1  .h  C
) )  -h  ( B  +h  ( -u 1  .h  D ) ) )
2922, 25, 283eqtr4i 2434 1  |-  ( ( A  -h  B )  -h  ( C  -h  D ) )  =  ( ( A  -h  C )  -h  ( B  -h  D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721  (class class class)co 6040   1c1 8947   -ucneg 9248   ~Hchil 22375    +h cva 22376    .h csm 22377    -h cmv 22381
This theorem is referenced by:  hvsubsub4  22515  pjsslem  23134
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-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hfvmul 22461  ax-hvdistr1 22464
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-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-ltxr 9081  df-sub 9249  df-neg 9250  df-hvsub 22427
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