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Theorem normpar 21736
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 5867 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21fveq2d 5531 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
32oveq1d 5875 . . . 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 5867 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) )
54fveq2d 5531 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) )
65oveq1d 5875 . . . 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 5878 . . 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 5527 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  A )  =  ( normh `  if ( A  e.  ~H ,  A ,  0h ) ) )
98oveq1d 5875 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  A ) ^ 2 )  =  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )
109oveq2d 5876 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
2  x.  ( (
normh `  A ) ^
2 ) )  =  ( 2  x.  (
( normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 ) ) )
1110oveq1d 5875 . . 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 2299 . 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 5868 . . . . . 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 5531 . . . . 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 5875 . . . 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 5868 . . . . . 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 5531 . . . . 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 5875 . . . 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 5878 . . 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 5527 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  B )  =  ( normh `  if ( B  e.  ~H ,  B ,  0h ) ) )
2120oveq1d 5875 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  B ) ^ 2 )  =  ( ( normh `  if ( B  e.  ~H ,  B ,  0h )
) ^ 2 ) )
2221oveq2d 5876 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
2  x.  ( (
normh `  B ) ^
2 ) )  =  ( 2  x.  (
( normh `  if ( B  e.  ~H ,  B ,  0h ) ) ^
2 ) ) )
2322oveq2d 5876 . . 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 2299 . 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 21585 . . . 4  |-  0h  e.  ~H
2625elimel 3619 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
2725elimel 3619 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
2826, 27normpari 21735 . 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 3609 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 1625    e. wcel 1686   ifcif 3567   ` cfv 5257  (class class class)co 5860    + caddc 8742    x. cmul 8744   2c2 9797   ^cexp 11106   ~Hchil 21501    +h cva 21502   normhcno 21505   0hc0v 21506    -h cmv 21507
This theorem is referenced by:  hhph  21759
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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