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Theorem normpari 21788
Description: Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normpar.1  |-  A  e. 
~H
normpar.2  |-  B  e. 
~H
Assertion
Ref Expression
normpari  |-  ( ( ( normh `  ( A  -h  B ) ) ^
2 )  +  ( ( normh `  ( A  +h  B ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  A
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B ) ^ 2 ) ) )

Proof of Theorem normpari
StepHypRef Expression
1 normpar.1 . . . . 5  |-  A  e. 
~H
2 normpar.2 . . . . 5  |-  B  e. 
~H
31, 2hvsubcli 21656 . . . 4  |-  ( A  -h  B )  e. 
~H
43normsqi 21766 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  =  ( ( A  -h  B
)  .ih  ( A  -h  B ) )
51, 2hvaddcli 21653 . . . 4  |-  ( A  +h  B )  e. 
~H
65normsqi 21766 . . 3  |-  ( (
normh `  ( A  +h  B ) ) ^
2 )  =  ( ( A  +h  B
)  .ih  ( A  +h  B ) )
74, 6oveq12i 5912 . 2  |-  ( ( ( normh `  ( A  -h  B ) ) ^
2 )  +  ( ( normh `  ( A  +h  B ) ) ^
2 ) )  =  ( ( ( A  -h  B )  .ih  ( A  -h  B
) )  +  ( ( A  +h  B
)  .ih  ( A  +h  B ) ) )
81normsqi 21766 . . . . . 6  |-  ( (
normh `  A ) ^
2 )  =  ( A  .ih  A )
98oveq2i 5911 . . . . 5  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( 2  x.  ( A 
.ih  A ) )
101, 1hicli 21715 . . . . . 6  |-  ( A 
.ih  A )  e.  CC
11102timesi 9892 . . . . 5  |-  ( 2  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
129, 11eqtri 2336 . . . 4  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
132normsqi 21766 . . . . . 6  |-  ( (
normh `  B ) ^
2 )  =  ( B  .ih  B )
1413oveq2i 5911 . . . . 5  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( 2  x.  ( B 
.ih  B ) )
152, 2hicli 21715 . . . . . 6  |-  ( B 
.ih  B )  e.  CC
16152timesi 9892 . . . . 5  |-  ( 2  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1714, 16eqtri 2336 . . . 4  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1812, 17oveq12i 5912 . . 3  |-  ( ( 2  x.  ( (
normh `  A ) ^
2 ) )  +  ( 2  x.  (
( normh `  B ) ^ 2 ) ) )  =  ( ( ( A  .ih  A
)  +  ( A 
.ih  A ) )  +  ( ( B 
.ih  B )  +  ( B  .ih  B
) ) )
191, 2, 1, 2normlem9 21752 . . . . . 6  |-  ( ( A  -h  B ) 
.ih  ( A  -h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  -  (
( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2010, 15addcli 8886 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  CC
211, 2hicli 21715 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
222, 1hicli 21715 . . . . . . . 8  |-  ( B 
.ih  A )  e.  CC
2321, 22addcli 8886 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  e.  CC
2420, 23negsubi 9169 . . . . . 6  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  -  (
( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2519, 24eqtr4i 2339 . . . . 5  |-  ( ( A  -h  B ) 
.ih  ( A  -h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  -u ( ( A  .ih  B )  +  ( B 
.ih  A ) ) )
261, 2, 1, 2normlem8 21751 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2725, 26oveq12i 5912 . . . 4  |-  ( ( ( A  -h  B
)  .ih  ( A  -h  B ) )  +  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  =  ( ( ( ( A  .ih  A )  +  ( B 
.ih  B ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  +  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) ) )
2823negcli 9159 . . . . 5  |-  -u (
( A  .ih  B
)  +  ( B 
.ih  A ) )  e.  CC
2920, 28, 20, 23add42i 9077 . . . 4  |-  ( ( ( ( A  .ih  A )  +  ( B 
.ih  B ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  +  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) ) )  =  ( ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  A
)  +  ( B 
.ih  B ) ) )  +  ( ( ( A  .ih  B
)  +  ( B 
.ih  A ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) ) )
3023negidi 9160 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( B 
.ih  A ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  0
3130oveq2i 5911 . . . . 5  |-  ( ( ( ( A  .ih  A )  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  +  ( ( ( A 
.ih  B )  +  ( B  .ih  A
) )  +  -u ( ( A  .ih  B )  +  ( B 
.ih  A ) ) ) )  =  ( ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  A
)  +  ( B 
.ih  B ) ) )  +  0 )
3220, 20addcli 8886 . . . . . 6  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  e.  CC
3332addid1i 9044 . . . . 5  |-  ( ( ( ( A  .ih  A )  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  +  0 )  =  ( ( ( A  .ih  A )  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )
3410, 15, 10, 15add4i 9076 . . . . 5  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  =  ( ( ( A 
.ih  A )  +  ( A  .ih  A
) )  +  ( ( B  .ih  B
)  +  ( B 
.ih  B ) ) )
3531, 33, 343eqtri 2340 . . . 4  |-  ( ( ( ( A  .ih  A )  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  +  ( ( ( A 
.ih  B )  +  ( B  .ih  A
) )  +  -u ( ( A  .ih  B )  +  ( B 
.ih  A ) ) ) )  =  ( ( ( A  .ih  A )  +  ( A 
.ih  A ) )  +  ( ( B 
.ih  B )  +  ( B  .ih  B
) ) )
3627, 29, 353eqtri 2340 . . 3  |-  ( ( ( A  -h  B
)  .ih  ( A  -h  B ) )  +  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )  =  ( ( ( A  .ih  A
)  +  ( A 
.ih  A ) )  +  ( ( B 
.ih  B )  +  ( B  .ih  B
) ) )
3718, 36eqtr4i 2339 . 2  |-  ( ( 2  x.  ( (
normh `  A ) ^
2 ) )  +  ( 2  x.  (
( normh `  B ) ^ 2 ) ) )  =  ( ( ( A  -h  B
)  .ih  ( A  -h  B ) )  +  ( ( A  +h  B )  .ih  ( A  +h  B ) ) )
387, 37eqtr4i 2339 1  |-  ( ( ( normh `  ( A  -h  B ) ) ^
2 )  +  ( ( normh `  ( A  +h  B ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  A
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B ) ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. wcel 1701   ` cfv 5292  (class class class)co 5900   0cc0 8782    + caddc 8785    x. cmul 8787    - cmin 9082   -ucneg 9083   2c2 9840   ^cexp 11151   ~Hchil 21554    +h cva 21555    .ih csp 21557   normhcno 21558    -h cmv 21560
This theorem is referenced by:  normpar  21789  normpar2i  21790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-hfvadd 21635  ax-hv0cl 21638  ax-hfvmul 21640  ax-hvmul0 21645  ax-hfi 21713  ax-his1 21716  ax-his2 21717  ax-his3 21718  ax-his4 21719
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-hnorm 21603  df-hvsub 21606
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