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Theorem normpari 21694
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 21562 . . . 4  |-  ( A  -h  B )  e. 
~H
43normsqi 21672 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  =  ( ( A  -h  B
)  .ih  ( A  -h  B ) )
51, 2hvaddcli 21559 . . . 4  |-  ( A  +h  B )  e. 
~H
65normsqi 21672 . . 3  |-  ( (
normh `  ( A  +h  B ) ) ^
2 )  =  ( ( A  +h  B
)  .ih  ( A  +h  B ) )
74, 6oveq12i 5804 . 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 21672 . . . . . 6  |-  ( (
normh `  A ) ^
2 )  =  ( A  .ih  A )
98oveq2i 5803 . . . . 5  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( 2  x.  ( A 
.ih  A ) )
101, 1hicli 21621 . . . . . 6  |-  ( A 
.ih  A )  e.  CC
11102timesi 9813 . . . . 5  |-  ( 2  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
129, 11eqtri 2278 . . . 4  |-  ( 2  x.  ( ( normh `  A ) ^ 2 ) )  =  ( ( A  .ih  A
)  +  ( A 
.ih  A ) )
132normsqi 21672 . . . . . 6  |-  ( (
normh `  B ) ^
2 )  =  ( B  .ih  B )
1413oveq2i 5803 . . . . 5  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( 2  x.  ( B 
.ih  B ) )
152, 2hicli 21621 . . . . . 6  |-  ( B 
.ih  B )  e.  CC
16152timesi 9813 . . . . 5  |-  ( 2  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1714, 16eqtri 2278 . . . 4  |-  ( 2  x.  ( ( normh `  B ) ^ 2 ) )  =  ( ( B  .ih  B
)  +  ( B 
.ih  B ) )
1812, 17oveq12i 5804 . . 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 21658 . . . . . 6  |-  ( ( A  -h  B ) 
.ih  ( A  -h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  -  (
( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2010, 15addcli 8809 . . . . . . 7  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  CC
211, 2hicli 21621 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
222, 1hicli 21621 . . . . . . . 8  |-  ( B 
.ih  A )  e.  CC
2321, 22addcli 8809 . . . . . . 7  |-  ( ( A  .ih  B )  +  ( B  .ih  A ) )  e.  CC
2420, 23negsubi 9092 . . . . . 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 2281 . . . . 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 21657 . . . . 5  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
2725, 26oveq12i 5804 . . . 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 9082 . . . . 5  |-  -u (
( A  .ih  B
)  +  ( B 
.ih  A ) )  e.  CC
2920, 28, 20, 23add42i 9000 . . . 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 9083 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( B 
.ih  A ) )  +  -u ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  0
3130oveq2i 5803 . . . . 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 8809 . . . . . 6  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  A )  +  ( B  .ih  B
) ) )  e.  CC
3332addid1i 8967 . . . . 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 8999 . . . . 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 2282 . . . 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 2282 . . 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 2281 . 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 2281 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 1619    e. wcel 1621   ` cfv 4673  (class class class)co 5792   0cc0 8705    + caddc 8708    x. cmul 8710    - cmin 9005   -ucneg 9006   2c2 9763   ^cexp 11071   ~Hchil 21460    +h cva 21461    .ih csp 21463   normhcno 21464    -h cmv 21466
This theorem is referenced by:  normpar  21695  normpar2i  21696
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-hfvadd 21541  ax-hv0cl 21544  ax-hfvmul 21546  ax-hvmul0 21551  ax-hfi 21619  ax-his1 21622  ax-his2 21623  ax-his3 21624  ax-his4 21625
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-seq 11014  df-exp 11072  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-hnorm 21509  df-hvsub 21512
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