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Theorem normpari 21729
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 21597 . . . 4  |-  ( A  -h  B )  e. 
~H
43normsqi 21707 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  =  ( ( A  -h  B
)  .ih  ( A  -h  B ) )
51, 2hvaddcli 21594 . . . 4  |-  ( A  +h  B )  e. 
~H
65normsqi 21707 . . 3  |-  ( (
normh `  ( A  +h  B ) ) ^
2 )  =  ( ( A  +h  B
)  .ih  ( A  +h  B ) )
74, 6oveq12i 5832 . 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 21707 . . . . . 6  |-  ( (
normh `  A ) ^
2 )  =  ( A  .ih  A )
98oveq2i 5831 . . . . 5  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( 2  x.  ( A 
.ih  A ) )
101, 1hicli 21656 . . . . . 6  |-  ( A 
.ih  A )  e.  CC
11102timesi 9841 . . . . 5  |-  ( 2  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
129, 11eqtri 2304 . . . 4  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
132normsqi 21707 . . . . . 6  |-  ( (
normh `  B ) ^
2 )  =  ( B  .ih  B )
1413oveq2i 5831 . . . . 5  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( 2  x.  ( B 
.ih  B ) )
152, 2hicli 21656 . . . . . 6  |-  ( B 
.ih  B )  e.  CC
16152timesi 9841 . . . . 5  |-  ( 2  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1714, 16eqtri 2304 . . . 4  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1812, 17oveq12i 5832 . . 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 21693 . . . . . 6  |-  ( ( A  -h  B ) 
.ih  ( A  -h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  -  (
( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2010, 15addcli 8837 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  CC
211, 2hicli 21656 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
222, 1hicli 21656 . . . . . . . 8  |-  ( B 
.ih  A )  e.  CC
2321, 22addcli 8837 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  e.  CC
2420, 23negsubi 9120 . . . . . 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 2307 . . . . 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 21692 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2725, 26oveq12i 5832 . . . 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 9110 . . . . 5  |-  -u (
( A  .ih  B
)  +  ( B 
.ih  A ) )  e.  CC
2920, 28, 20, 23add42i 9028 . . . 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 9111 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( B 
.ih  A ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  0
3130oveq2i 5831 . . . . 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 8837 . . . . . 6  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  e.  CC
3332addid1i 8995 . . . . 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 9027 . . . . 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 2308 . . . 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 2308 . . 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 2307 . 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 2307 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 1623    e. wcel 1685   ` cfv 5221  (class class class)co 5820   0cc0 8733    + caddc 8736    x. cmul 8738    - cmin 9033   -ucneg 9034   2c2 9791   ^cexp 11100   ~Hchil 21495    +h cva 21496    .ih csp 21498   normhcno 21499    -h cmv 21501
This theorem is referenced by:  normpar  21730  normpar2i  21731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-hfvadd 21576  ax-hv0cl 21579  ax-hfvmul 21581  ax-hvmul0 21586  ax-hfi 21654  ax-his1 21657  ax-his2 21658  ax-his3 21659  ax-his4 21660
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-hnorm 21544  df-hvsub 21547
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