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Theorem normlem8 22612
Description: Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem8.1  |-  A  e. 
~H
normlem8.2  |-  B  e. 
~H
normlem8.3  |-  C  e. 
~H
normlem8.4  |-  D  e. 
~H
Assertion
Ref Expression
normlem8  |-  ( ( A  +h  B ) 
.ih  ( C  +h  D ) )  =  ( ( ( A 
.ih  C )  +  ( B  .ih  D
) )  +  ( ( A  .ih  D
)  +  ( B 
.ih  C ) ) )

Proof of Theorem normlem8
StepHypRef Expression
1 normlem8.1 . . . 4  |-  A  e. 
~H
2 normlem8.3 . . . 4  |-  C  e. 
~H
3 normlem8.4 . . . 4  |-  D  e. 
~H
4 his7 22585 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H  /\  D  e.  ~H )  ->  ( A  .ih  ( C  +h  D ) )  =  ( ( A  .ih  C )  +  ( A 
.ih  D ) ) )
51, 2, 3, 4mp3an 1279 . . 3  |-  ( A 
.ih  ( C  +h  D ) )  =  ( ( A  .ih  C )  +  ( A 
.ih  D ) )
6 normlem8.2 . . . 4  |-  B  e. 
~H
7 his7 22585 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  D  e.  ~H )  ->  ( B  .ih  ( C  +h  D ) )  =  ( ( B  .ih  C )  +  ( B 
.ih  D ) ) )
86, 2, 3, 7mp3an 1279 . . 3  |-  ( B 
.ih  ( C  +h  D ) )  =  ( ( B  .ih  C )  +  ( B 
.ih  D ) )
95, 8oveq12i 6086 . 2  |-  ( ( A  .ih  ( C  +h  D ) )  +  ( B  .ih  ( C  +h  D
) ) )  =  ( ( ( A 
.ih  C )  +  ( A  .ih  D
) )  +  ( ( B  .ih  C
)  +  ( B 
.ih  D ) ) )
102, 3hvaddcli 22514 . . 3  |-  ( C  +h  D )  e. 
~H
11 ax-his2 22578 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  ( C  +h  D )  e. 
~H )  ->  (
( A  +h  B
)  .ih  ( C  +h  D ) )  =  ( ( A  .ih  ( C  +h  D
) )  +  ( B  .ih  ( C  +h  D ) ) ) )
121, 6, 10, 11mp3an 1279 . 2  |-  ( ( A  +h  B ) 
.ih  ( C  +h  D ) )  =  ( ( A  .ih  ( C  +h  D
) )  +  ( B  .ih  ( C  +h  D ) ) )
131, 2hicli 22576 . . 3  |-  ( A 
.ih  C )  e.  CC
146, 3hicli 22576 . . 3  |-  ( B 
.ih  D )  e.  CC
151, 3hicli 22576 . . 3  |-  ( A 
.ih  D )  e.  CC
166, 2hicli 22576 . . 3  |-  ( B 
.ih  C )  e.  CC
1713, 14, 15, 16add42i 9279 . 2  |-  ( ( ( A  .ih  C
)  +  ( B 
.ih  D ) )  +  ( ( A 
.ih  D )  +  ( B  .ih  C
) ) )  =  ( ( ( A 
.ih  C )  +  ( A  .ih  D
) )  +  ( ( B  .ih  C
)  +  ( B 
.ih  D ) ) )
189, 12, 173eqtr4i 2466 1  |-  ( ( A  +h  B ) 
.ih  ( C  +h  D ) )  =  ( ( ( A 
.ih  C )  +  ( B  .ih  D
) )  +  ( ( A  .ih  D
)  +  ( B 
.ih  C ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725  (class class class)co 6074    + caddc 8986   ~Hchil 22415    +h cva 22416    .ih csp 22418
This theorem is referenced by:  normlem9  22613  norm-ii-i  22632  normpythi  22637  normpari  22649  polid2i  22652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-hfvadd 22496  ax-hfi 22574  ax-his1 22577  ax-his2 22578
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-po 4496  df-so 4497  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-riota 6542  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-2 10051  df-cj 11897  df-re 11898  df-im 11899
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