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Theorem normpar 21750
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 5881 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21fveq2d 5545 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
32oveq1d 5889 . . . 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 5881 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) )
54fveq2d 5545 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) )
65oveq1d 5889 . . . 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 5892 . . 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 5541 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  A )  =  ( normh `  if ( A  e.  ~H ,  A ,  0h ) ) )
98oveq1d 5889 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  A ) ^ 2 )  =  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )
109oveq2d 5890 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
2  x.  ( (
normh `  A ) ^
2 ) )  =  ( 2  x.  (
( normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 ) ) )
1110oveq1d 5889 . . 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 2310 . 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 5882 . . . . . 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 5545 . . . . 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 5889 . . . 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 5882 . . . . . 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 5545 . . . . 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 5889 . . . 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 5892 . . 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 5541 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  B )  =  ( normh `  if ( B  e.  ~H ,  B ,  0h ) ) )
2120oveq1d 5889 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  B ) ^ 2 )  =  ( ( normh `  if ( B  e.  ~H ,  B ,  0h )
) ^ 2 ) )
2221oveq2d 5890 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
2  x.  ( (
normh `  B ) ^
2 ) )  =  ( 2  x.  (
( normh `  if ( B  e.  ~H ,  B ,  0h ) ) ^
2 ) ) )
2322oveq2d 5890 . . 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 2310 . 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 21599 . . . 4  |-  0h  e.  ~H
2625elimel 3630 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
2725elimel 3630 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
2826, 27normpari 21749 . 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 3620 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 358    = wceq 1632    e. wcel 1696   ifcif 3578   ` cfv 5271  (class class class)co 5874    + caddc 8756    x. cmul 8758   2c2 9811   ^cexp 11120   ~Hchil 21515    +h cva 21516   normhcno 21519   0hc0v 21520    -h cmv 21521
This theorem is referenced by:  hhph  21773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-hfvadd 21596  ax-hv0cl 21599  ax-hfvmul 21601  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-hnorm 21564  df-hvsub 21567
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