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Theorem normpar 22649
Description: Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
normpar  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( normh `  ( A  -h  B
) ) ^ 2 )  +  ( (
normh `  ( A  +h  B ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  A
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B ) ^ 2 ) ) ) )

Proof of Theorem normpar
StepHypRef Expression
1 oveq1 6080 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21fveq2d 5724 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
32oveq1d 6088 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  -h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )
4 oveq1 6080 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) )
54fveq2d 5724 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) )
65oveq1d 6088 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  +h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 ) )
73, 6oveq12d 6091 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( normh `  ( A  -h  B ) ) ^ 2 )  +  ( ( normh `  ( A  +h  B ) ) ^ 2 ) )  =  ( ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 )  +  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 ) ) )
8 fveq2 5720 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  A )  =  ( normh `  if ( A  e.  ~H ,  A ,  0h ) ) )
98oveq1d 6088 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  A ) ^ 2 )  =  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )
109oveq2d 6089 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
2  x.  ( (
normh `  A ) ^
2 ) )  =  ( 2  x.  (
( normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 ) ) )
1110oveq1d 6088 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( 2  x.  (
( normh `  A ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B
) ^ 2 ) ) )  =  ( ( 2  x.  (
( normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  B ) ^ 2 ) ) ) )
127, 11eqeq12d 2449 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( ( normh `  ( A  -h  B
) ) ^ 2 )  +  ( (
normh `  ( A  +h  B ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  A
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B ) ^ 2 ) ) )  <->  ( (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 )  +  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B ) ^ 2 ) ) ) ) )
13 oveq2 6081 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )
1413fveq2d 5724 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1514oveq1d 6088 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )
16 oveq2 6081 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  +h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
) )
1716fveq2d 5724 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1817oveq1d 6088 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )
1915, 18oveq12d 6091 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 )  +  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 ) )  =  ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) ) ^ 2 )  +  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) ) )
20 fveq2 5720 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  B )  =  ( normh `  if ( B  e.  ~H ,  B ,  0h ) ) )
2120oveq1d 6088 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  B ) ^ 2 )  =  ( ( normh `  if ( B  e.  ~H ,  B ,  0h )
) ^ 2 ) )
2221oveq2d 6089 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
2  x.  ( (
normh `  B ) ^
2 ) )  =  ( 2  x.  (
( normh `  if ( B  e.  ~H ,  B ,  0h ) ) ^
2 ) ) )
2322oveq2d 6089 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( 2  x.  (
( normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  B ) ^ 2 ) ) )  =  ( ( 2  x.  ( (
normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  if ( B  e.  ~H ,  B ,  0h ) ) ^
2 ) ) ) )
2419, 23eqeq12d 2449 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) ) ^ 2 )  +  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B ) ^ 2 ) ) )  <->  ( (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 )  +  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  if ( B  e. 
~H ,  B ,  0h ) ) ^ 2 ) ) ) ) )
25 ax-hv0cl 22498 . . . 4  |-  0h  e.  ~H
2625elimel 3783 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
2725elimel 3783 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
2826, 27normpari 22648 . 2  |-  ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 )  +  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  if ( B  e. 
~H ,  B ,  0h ) ) ^ 2 ) ) )
2912, 24, 28dedth2h 3773 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( 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:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3731   ` cfv 5446  (class class class)co 6073    + caddc 8985    x. cmul 8987   2c2 10041   ^cexp 11374   ~Hchil 22414    +h cva 22415   normhcno 22418   0hc0v 22419    -h cmv 22420
This theorem is referenced by:  hhph  22672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-hfvadd 22495  ax-hv0cl 22498  ax-hfvmul 22500  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-hnorm 22463  df-hvsub 22466
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