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Theorem normpari 22617
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 22485 . . . 4  |-  ( A  -h  B )  e. 
~H
43normsqi 22595 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  =  ( ( A  -h  B
)  .ih  ( A  -h  B ) )
51, 2hvaddcli 22482 . . . 4  |-  ( A  +h  B )  e. 
~H
65normsqi 22595 . . 3  |-  ( (
normh `  ( A  +h  B ) ) ^
2 )  =  ( ( A  +h  B
)  .ih  ( A  +h  B ) )
74, 6oveq12i 6060 . 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 22595 . . . . . 6  |-  ( (
normh `  A ) ^
2 )  =  ( A  .ih  A )
98oveq2i 6059 . . . . 5  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( 2  x.  ( A 
.ih  A ) )
101, 1hicli 22544 . . . . . 6  |-  ( A 
.ih  A )  e.  CC
11102timesi 10065 . . . . 5  |-  ( 2  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
129, 11eqtri 2432 . . . 4  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
132normsqi 22595 . . . . . 6  |-  ( (
normh `  B ) ^
2 )  =  ( B  .ih  B )
1413oveq2i 6059 . . . . 5  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( 2  x.  ( B 
.ih  B ) )
152, 2hicli 22544 . . . . . 6  |-  ( B 
.ih  B )  e.  CC
16152timesi 10065 . . . . 5  |-  ( 2  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1714, 16eqtri 2432 . . . 4  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1812, 17oveq12i 6060 . . 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 22581 . . . . . 6  |-  ( ( A  -h  B ) 
.ih  ( A  -h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  -  (
( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2010, 15addcli 9058 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  CC
211, 2hicli 22544 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
222, 1hicli 22544 . . . . . . . 8  |-  ( B 
.ih  A )  e.  CC
2321, 22addcli 9058 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  e.  CC
2420, 23negsubi 9342 . . . . . 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 2435 . . . . 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 22580 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2725, 26oveq12i 6060 . . . 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 9332 . . . . 5  |-  -u (
( A  .ih  B
)  +  ( B 
.ih  A ) )  e.  CC
2920, 28, 20, 23add42i 9250 . . . 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 9333 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( B 
.ih  A ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  0
3130oveq2i 6059 . . . . 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 9058 . . . . . 6  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  e.  CC
3332addid1i 9217 . . . . 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 9249 . . . . 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 2436 . . . 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 2436 . . 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 2435 . 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 2435 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 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   0cc0 8954    + caddc 8957    x. cmul 8959    - cmin 9255   -ucneg 9256   2c2 10013   ^cexp 11345   ~Hchil 22383    +h cva 22384    .ih csp 22386   normhcno 22387    -h cmv 22389
This theorem is referenced by:  normpar  22618  normpar2i  22619
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-hfvadd 22464  ax-hv0cl 22467  ax-hfvmul 22469  ax-hvmul0 22474  ax-hfi 22542  ax-his1 22545  ax-his2 22546  ax-his3 22547  ax-his4 22548
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-hnorm 22432  df-hvsub 22435
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