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Theorem normpari 21735
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 21603 . . . 4  |-  ( A  -h  B )  e. 
~H
43normsqi 21713 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  =  ( ( A  -h  B
)  .ih  ( A  -h  B ) )
51, 2hvaddcli 21600 . . . 4  |-  ( A  +h  B )  e. 
~H
65normsqi 21713 . . 3  |-  ( (
normh `  ( A  +h  B ) ) ^
2 )  =  ( ( A  +h  B
)  .ih  ( A  +h  B ) )
74, 6oveq12i 5872 . 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 21713 . . . . . 6  |-  ( (
normh `  A ) ^
2 )  =  ( A  .ih  A )
98oveq2i 5871 . . . . 5  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( 2  x.  ( A 
.ih  A ) )
101, 1hicli 21662 . . . . . 6  |-  ( A 
.ih  A )  e.  CC
11102timesi 9847 . . . . 5  |-  ( 2  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
129, 11eqtri 2305 . . . 4  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
132normsqi 21713 . . . . . 6  |-  ( (
normh `  B ) ^
2 )  =  ( B  .ih  B )
1413oveq2i 5871 . . . . 5  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( 2  x.  ( B 
.ih  B ) )
152, 2hicli 21662 . . . . . 6  |-  ( B 
.ih  B )  e.  CC
16152timesi 9847 . . . . 5  |-  ( 2  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1714, 16eqtri 2305 . . . 4  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1812, 17oveq12i 5872 . . 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 21699 . . . . . 6  |-  ( ( A  -h  B ) 
.ih  ( A  -h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  -  (
( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2010, 15addcli 8843 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  CC
211, 2hicli 21662 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
222, 1hicli 21662 . . . . . . . 8  |-  ( B 
.ih  A )  e.  CC
2321, 22addcli 8843 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  e.  CC
2420, 23negsubi 9126 . . . . . 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 2308 . . . . 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 21698 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2725, 26oveq12i 5872 . . . 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 9116 . . . . 5  |-  -u (
( A  .ih  B
)  +  ( B 
.ih  A ) )  e.  CC
2920, 28, 20, 23add42i 9034 . . . 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 9117 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( B 
.ih  A ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  0
3130oveq2i 5871 . . . . 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 8843 . . . . . 6  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  e.  CC
3332addid1i 9001 . . . . 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 9033 . . . . 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 2309 . . . 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 2309 . . 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 2308 . 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 2308 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 1625    e. wcel 1686   ` cfv 5257  (class class class)co 5860   0cc0 8739    + caddc 8742    x. cmul 8744    - cmin 9039   -ucneg 9040   2c2 9797   ^cexp 11106   ~Hchil 21501    +h cva 21502    .ih csp 21504   normhcno 21505    -h cmv 21507
This theorem is referenced by:  normpar  21736  normpar2i  21737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-hfvadd 21582  ax-hv0cl 21585  ax-hfvmul 21587  ax-hvmul0 21592  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665  ax-his4 21666
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-sup 7196  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-hnorm 21550  df-hvsub 21553
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