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Theorem normlem8 22460
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 22433 . . . 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 22433 . . . 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 6025 . 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 22362 . . 3  |-  ( C  +h  D )  e. 
~H
11 ax-his2 22426 . . 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 22424 . . 3  |-  ( A 
.ih  C )  e.  CC
146, 3hicli 22424 . . 3  |-  ( B 
.ih  D )  e.  CC
151, 3hicli 22424 . . 3  |-  ( A 
.ih  D )  e.  CC
166, 2hicli 22424 . . 3  |-  ( B 
.ih  C )  e.  CC
1713, 14, 15, 16add42i 9211 . 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 2410 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 1717  (class class class)co 6013    + caddc 8919   ~Hchil 22263    +h cva 22264    .ih csp 22266
This theorem is referenced by:  normlem9  22461  norm-ii-i  22480  normpythi  22485  normpari  22497  polid2i  22500
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-hfvadd 22344  ax-hfi 22422  ax-his1 22425  ax-his2 22426
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983  df-cj 11824  df-re 11825  df-im 11826
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