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Theorem normpar2i 21751
Description: Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normpar2.1  |-  A  e. 
~H
normpar2.2  |-  B  e. 
~H
normpar2.3  |-  C  e. 
~H
Assertion
Ref Expression
normpar2i  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  =  ( ( ( 2  x.  ( ( normh `  ( A  -h  C ) ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  ( B  -h  C ) ) ^ 2 ) ) )  -  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 ) )

Proof of Theorem normpar2i
StepHypRef Expression
1 2re 9831 . . . . . 6  |-  2  e.  RR
2 normpar2.1 . . . . . . . . 9  |-  A  e. 
~H
3 normpar2.3 . . . . . . . . 9  |-  C  e. 
~H
42, 3hvsubcli 21617 . . . . . . . 8  |-  ( A  -h  C )  e. 
~H
54normcli 21726 . . . . . . 7  |-  ( normh `  ( A  -h  C
) )  e.  RR
65resqcli 11205 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  RR
71, 6remulcli 8867 . . . . 5  |-  ( 2  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  RR
8 normpar2.2 . . . . . . . . 9  |-  B  e. 
~H
98, 3hvsubcli 21617 . . . . . . . 8  |-  ( B  -h  C )  e. 
~H
109normcli 21726 . . . . . . 7  |-  ( normh `  ( B  -h  C
) )  e.  RR
1110resqcli 11205 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  RR
121, 11remulcli 8867 . . . . 5  |-  ( 2  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  RR
137, 12readdcli 8866 . . . 4  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  RR
1413recni 8865 . . 3  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  CC
152, 8hvaddcli 21614 . . . . . . 7  |-  ( A  +h  B )  e. 
~H
16 2cn 9832 . . . . . . . 8  |-  2  e.  CC
1716, 3hvmulcli 21610 . . . . . . 7  |-  ( 2  .h  C )  e. 
~H
1815, 17hvsubcli 21617 . . . . . 6  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  e. 
~H
1918normcli 21726 . . . . 5  |-  ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) )  e.  RR
2019resqcli 11205 . . . 4  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  RR
2120recni 8865 . . 3  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  CC
222, 8hvsubcli 21617 . . . . . 6  |-  ( A  -h  B )  e. 
~H
2322normcli 21726 . . . . 5  |-  ( normh `  ( A  -h  B
) )  e.  RR
2423resqcli 11205 . . . 4  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  RR
2524recni 8865 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  CC
26 4cn 9836 . . . . . 6  |-  4  e.  CC
276recni 8865 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  CC
2826, 27mulcli 8858 . . . . 5  |-  ( 4  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  CC
2911recni 8865 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  CC
3026, 29mulcli 8858 . . . . 5  |-  ( 4  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  CC
31 2ne0 9845 . . . . 5  |-  2  =/=  0
3228, 30, 16, 31divdiri 9533 . . . 4  |-  ( ( ( 4  x.  (
( normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 4  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  /  2 )  =  ( ( ( 4  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  /  2
)  +  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 ) )
3328, 30addcomi 9019 . . . . . . . 8  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 4  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  =  ( ( 4  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  +  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) ) )
34 neg1cn 9829 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
3534, 17hvmulcli 21610 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( 2  .h  C ) )  e.  ~H
3634, 22hvmulcli 21610 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( A  -h  B ) )  e.  ~H
3715, 35, 36hvadd32i 21649 . . . . . . . . . . . . . . 15  |-  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( 2  .h  C ) ) )  +h  ( -u
1  .h  ( A  -h  B ) ) )  =  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( A  -h  B ) ) )  +h  ( -u
1  .h  ( 2  .h  C ) ) )
3815, 17hvsubvali 21616 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  =  ( ( A  +h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
3938oveq1i 5884 . . . . . . . . . . . . . . 15  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( ( A  +h  B )  +h  ( -u 1  .h  ( 2  .h  C
) ) )  +h  ( -u 1  .h  ( A  -h  B
) ) )
4016, 8hvmulcli 21610 . . . . . . . . . . . . . . . . 17  |-  ( 2  .h  B )  e. 
~H
4140, 17hvsubvali 21616 . . . . . . . . . . . . . . . 16  |-  ( ( 2  .h  B )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
422, 8hvcomi 21615 . . . . . . . . . . . . . . . . . . 19  |-  ( A  +h  B )  =  ( B  +h  A
)
432, 8hvnegdii 21657 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1  .h  ( A  -h  B ) )  =  ( B  -h  A )
4442, 43oveq12i 5886 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( B  +h  A )  +h  ( B  -h  A ) )
458, 2hvsubcan2i 21659 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  +h  A )  +h  ( B  -h  A ) )  =  ( 2  .h  B
)
4644, 45eqtri 2316 . . . . . . . . . . . . . . . . 17  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( 2  .h  B
)
4746oveq1i 5884 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( A  -h  B ) ) )  +h  ( -u
1  .h  ( 2  .h  C ) ) )  =  ( ( 2  .h  B )  +h  ( -u 1  .h  ( 2  .h  C
) ) )
4841, 47eqtr4i 2319 . . . . . . . . . . . . . . 15  |-  ( ( 2  .h  B )  -h  ( 2  .h  C ) )  =  ( ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  +h  ( -u 1  .h  ( 2  .h  C
) ) )
4937, 39, 483eqtr4i 2326 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5018, 22hvsubvali 21616 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  -h  ( A  -h  B ) )  =  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( -u 1  .h  ( A  -h  B ) ) )
5116, 8, 3hvsubdistr1i 21647 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( B  -h  C ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5249, 50, 513eqtr4i 2326 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  -h  ( A  -h  B ) )  =  ( 2  .h  ( B  -h  C ) )
5352fveq2i 5544 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( B  -h  C ) ) )
5416, 9norm-iii-i 21734 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( B  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )
55 0re 8854 . . . . . . . . . . . . . . 15  |-  0  e.  RR
56 2pos 9844 . . . . . . . . . . . . . . 15  |-  0  <  2
5755, 1, 56ltleii 8957 . . . . . . . . . . . . . 14  |-  0  <_  2
581absidi 11877 . . . . . . . . . . . . . 14  |-  ( 0  <_  2  ->  ( abs `  2 )  =  2 )
5957, 58ax-mp 8 . . . . . . . . . . . . 13  |-  ( abs `  2 )  =  2
6059oveq1i 5884 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6153, 54, 603eqtri 2320 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6261oveq1i 5884 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( B  -h  C ) ) ) ^ 2 )
6310recni 8865 . . . . . . . . . . 11  |-  ( normh `  ( B  -h  C
) )  e.  CC
6416, 63sqmuli 11203 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( B  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
65 sq2 11215 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
6665oveq1i 5884 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
6762, 64, 663eqtri 2320 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
682, 8hvsubcan2i 21659 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  +h  ( A  -h  B ) )  =  ( 2  .h  A
)
6968oveq1i 5884 . . . . . . . . . . . . . . 15  |-  ( ( ( A  +h  B
)  +h  ( A  -h  B ) )  +h  ( -u 1  .h  ( 2  .h  C
) ) )  =  ( ( 2  .h  A )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
7015, 35, 22hvadd32i 21649 . . . . . . . . . . . . . . 15  |-  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( 2  .h  C ) ) )  +h  ( A  -h  B ) )  =  ( ( ( A  +h  B )  +h  ( A  -h  B ) )  +h  ( -u 1  .h  ( 2  .h  C
) ) )
7116, 2hvmulcli 21610 . . . . . . . . . . . . . . . 16  |-  ( 2  .h  A )  e. 
~H
7271, 17hvsubvali 21616 . . . . . . . . . . . . . . 15  |-  ( ( 2  .h  A )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  A )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
7369, 70, 723eqtr4i 2326 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( 2  .h  C ) ) )  +h  ( A  -h  B ) )  =  ( ( 2  .h  A )  -h  ( 2  .h  C
) )
7438oveq1i 5884 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( A  -h  B ) )  =  ( ( ( A  +h  B )  +h  ( -u 1  .h  ( 2  .h  C
) ) )  +h  ( A  -h  B
) )
7516, 2, 3hvsubdistr1i 21647 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( A  -h  C ) )  =  ( ( 2  .h  A )  -h  (
2  .h  C ) )
7673, 74, 753eqtr4i 2326 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( A  -h  B ) )  =  ( 2  .h  ( A  -h  C ) )
7776fveq2i 5544 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( A  -h  C ) ) )
7816, 4norm-iii-i 21734 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( A  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )
7959oveq1i 5884 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8077, 78, 793eqtri 2320 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8180oveq1i 5884 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( A  -h  C ) ) ) ^ 2 )
825recni 8865 . . . . . . . . . . 11  |-  ( normh `  ( A  -h  C
) )  e.  CC
8316, 82sqmuli 11203 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( A  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8465oveq1i 5884 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8581, 83, 843eqtri 2320 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8667, 85oveq12i 5886 . . . . . . . 8  |-  ( ( ( normh `  ( (
( A  +h  B
)  -h  ( 2  .h  C ) )  -h  ( A  -h  B ) ) ) ^ 2 )  +  ( ( normh `  (
( ( A  +h  B )  -h  (
2  .h  C ) )  +h  ( A  -h  B ) ) ) ^ 2 ) )  =  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  +  ( 4  x.  (
( normh `  ( A  -h  C ) ) ^
2 ) ) )
8733, 86eqtr4i 2319 . . . . . . 7  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 4  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  =  ( ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  +  ( ( normh `  ( (
( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( A  -h  B ) ) ) ^ 2 ) )
8818, 22normpari 21749 . . . . . . 7  |-  ( ( ( normh `  ( (
( A  +h  B
)  -h  ( 2  .h  C ) )  -h  ( A  -h  B ) ) ) ^ 2 )  +  ( ( normh `  (
( ( A  +h  B )  -h  (
2  .h  C ) )  +h  ( A  -h  B ) ) ) ^ 2 ) )  =  ( ( 2  x.  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( A  -h  B ) ) ^
2 ) ) )
8987, 88eqtri 2316 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 4  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  =  ( ( 2  x.  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  +  ( 2  x.  ( (
normh `  ( A  -h  B ) ) ^
2 ) ) )
9089oveq1i 5884 . . . . 5  |-  ( ( ( 4  x.  (
( normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 4  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  /  2 )  =  ( ( ( 2  x.  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  +  ( 2  x.  ( (
normh `  ( A  -h  B ) ) ^
2 ) ) )  /  2 )
9116, 21mulcli 8858 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  e.  CC
9216, 25mulcli 8858 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( A  -h  B
) ) ^ 2 ) )  e.  CC
9391, 92, 16, 31divdiri 9533 . . . . 5  |-  ( ( ( 2  x.  (
( normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( A  -h  B ) ) ^
2 ) ) )  /  2 )  =  ( ( ( 2  x.  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  /  2
)  +  ( ( 2  x.  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  / 
2 ) )
9421, 16, 31divcan3i 9522 . . . . . 6  |-  ( ( 2  x.  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 ) )  / 
2 )  =  ( ( normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )
9525, 16, 31divcan3i 9522 . . . . . 6  |-  ( ( 2  x.  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  / 
2 )  =  ( ( normh `  ( A  -h  B ) ) ^
2 )
9694, 95oveq12i 5886 . . . . 5  |-  ( ( ( 2  x.  (
( normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 ) )  / 
2 )  +  ( ( 2  x.  (
( normh `  ( A  -h  B ) ) ^
2 ) )  / 
2 ) )  =  ( ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 )  +  ( (
normh `  ( A  -h  B ) ) ^
2 ) )
9790, 93, 963eqtri 2320 . . . 4  |-  ( ( ( 4  x.  (
( normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 4  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  /  2 )  =  ( ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 )  +  ( (
normh `  ( A  -h  B ) ) ^
2 ) )
9826, 27, 16, 31div23i 9534 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
99 4d2e2 9892 . . . . . . 7  |-  ( 4  /  2 )  =  2
10099oveq1i 5884 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10198, 100eqtri 2316 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10226, 29, 16, 31div23i 9534 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
10399oveq1i 5884 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
104102, 103eqtri 2316 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
105101, 104oveq12i 5886 . . . 4  |-  ( ( ( 4  x.  (
( normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  +  ( ( 4  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 ) )  =  ( ( 2  x.  ( ( normh `  ( A  -h  C ) ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  ( B  -h  C ) ) ^ 2 ) ) )
10632, 97, 1053eqtr3i 2324 . . 3  |-  ( ( ( normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  +  ( ( normh `  ( A  -h  B ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  ( A  -h  C ) ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  ( B  -h  C ) ) ^ 2 ) ) )
10714, 21, 25, 106subaddrii 9151 . 2  |-  ( ( ( 2  x.  (
( normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  -  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  =  ( ( normh `  ( A  -h  B ) ) ^
2 )
108107eqcomi 2300 1  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  =  ( ( ( 2  x.  ( ( normh `  ( A  -h  C ) ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  ( B  -h  C ) ) ^ 2 ) ) )  -  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   4c4 9813   ^cexp 11120   abscabs 11735   ~Hchil 21515    +h cva 21516    .h csm 21517   normhcno 21519    -h cmv 21521
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-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  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-4 9822  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-abs 11737  df-hnorm 21564  df-hvsub 21567
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