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Theorem normlem8 22572
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 22545 . . . 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 22545 . . . 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 6052 . 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 22474 . . 3  |-  ( C  +h  D )  e. 
~H
11 ax-his2 22538 . . 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 22536 . . 3  |-  ( A 
.ih  C )  e.  CC
146, 3hicli 22536 . . 3  |-  ( B 
.ih  D )  e.  CC
151, 3hicli 22536 . . 3  |-  ( A 
.ih  D )  e.  CC
166, 2hicli 22536 . . 3  |-  ( B 
.ih  C )  e.  CC
1713, 14, 15, 16add42i 9242 . 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 2434 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 1649    e. wcel 1721  (class class class)co 6040    + caddc 8949   ~Hchil 22375    +h cva 22376    .ih csp 22378
This theorem is referenced by:  normlem9  22573  norm-ii-i  22592  normpythi  22597  normpari  22609  polid2i  22612
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-hfi 22534  ax-his1 22537  ax-his2 22538
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-2 10014  df-cj 11859  df-re 11860  df-im 11861
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