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Theorem normpar2i 21728
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 9811 . . . . . 6  |-  2  e.  RR
2 normpar2.1 . . . . . . . . 9  |-  A  e. 
~H
3 normpar2.3 . . . . . . . . 9  |-  C  e. 
~H
42, 3hvsubcli 21594 . . . . . . . 8  |-  ( A  -h  C )  e. 
~H
54normcli 21703 . . . . . . 7  |-  ( normh `  ( A  -h  C
) )  e.  RR
65resqcli 11184 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  RR
71, 6remulcli 8847 . . . . 5  |-  ( 2  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  RR
8 normpar2.2 . . . . . . . . 9  |-  B  e. 
~H
98, 3hvsubcli 21594 . . . . . . . 8  |-  ( B  -h  C )  e. 
~H
109normcli 21703 . . . . . . 7  |-  ( normh `  ( B  -h  C
) )  e.  RR
1110resqcli 11184 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  RR
121, 11remulcli 8847 . . . . 5  |-  ( 2  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  RR
137, 12readdcli 8846 . . . 4  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  RR
1413recni 8845 . . 3  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  CC
152, 8hvaddcli 21591 . . . . . . 7  |-  ( A  +h  B )  e. 
~H
16 2cn 9812 . . . . . . . 8  |-  2  e.  CC
1716, 3hvmulcli 21587 . . . . . . 7  |-  ( 2  .h  C )  e. 
~H
1815, 17hvsubcli 21594 . . . . . 6  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  e. 
~H
1918normcli 21703 . . . . 5  |-  ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) )  e.  RR
2019resqcli 11184 . . . 4  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  RR
2120recni 8845 . . 3  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  CC
222, 8hvsubcli 21594 . . . . . 6  |-  ( A  -h  B )  e. 
~H
2322normcli 21703 . . . . 5  |-  ( normh `  ( A  -h  B
) )  e.  RR
2423resqcli 11184 . . . 4  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  RR
2524recni 8845 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  CC
26 4cn 9816 . . . . . 6  |-  4  e.  CC
276recni 8845 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  CC
2826, 27mulcli 8838 . . . . 5  |-  ( 4  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  CC
2911recni 8845 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  CC
3026, 29mulcli 8838 . . . . 5  |-  ( 4  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  CC
31 2ne0 9825 . . . . 5  |-  2  =/=  0
3228, 30, 16, 31divdiri 9513 . . . 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 8999 . . . . . . . 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 9809 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
3534, 17hvmulcli 21587 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( 2  .h  C ) )  e.  ~H
3634, 22hvmulcli 21587 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( A  -h  B ) )  e.  ~H
3715, 35, 36hvadd32i 21626 . . . . . . . . . . . . . . 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 21593 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  =  ( ( A  +h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
3938oveq1i 5830 . . . . . . . . . . . . . . 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 21587 . . . . . . . . . . . . . . . . 17  |-  ( 2  .h  B )  e. 
~H
4140, 17hvsubvali 21593 . . . . . . . . . . . . . . . 16  |-  ( ( 2  .h  B )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
422, 8hvcomi 21592 . . . . . . . . . . . . . . . . . . 19  |-  ( A  +h  B )  =  ( B  +h  A
)
432, 8hvnegdii 21634 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1  .h  ( A  -h  B ) )  =  ( B  -h  A )
4442, 43oveq12i 5832 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( B  +h  A )  +h  ( B  -h  A ) )
458, 2hvsubcan2i 21636 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  +h  A )  +h  ( B  -h  A ) )  =  ( 2  .h  B
)
4644, 45eqtri 2305 . . . . . . . . . . . . . . . . 17  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( 2  .h  B
)
4746oveq1i 5830 . . . . . . . . . . . . . . . 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 2308 . . . . . . . . . . . . . . 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 2315 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5018, 22hvsubvali 21593 . . . . . . . . . . . . . 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 21624 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( B  -h  C ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5249, 50, 513eqtr4i 2315 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  -h  ( A  -h  B ) )  =  ( 2  .h  ( B  -h  C ) )
5352fveq2i 5489 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( B  -h  C ) ) )
5416, 9norm-iii-i 21711 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( B  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )
55 0re 8834 . . . . . . . . . . . . . . 15  |-  0  e.  RR
56 2pos 9824 . . . . . . . . . . . . . . 15  |-  0  <  2
5755, 1, 56ltleii 8937 . . . . . . . . . . . . . 14  |-  0  <_  2
581absidi 11856 . . . . . . . . . . . . . 14  |-  ( 0  <_  2  ->  ( abs `  2 )  =  2 )
5957, 58ax-mp 10 . . . . . . . . . . . . 13  |-  ( abs `  2 )  =  2
6059oveq1i 5830 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6153, 54, 603eqtri 2309 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6261oveq1i 5830 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( B  -h  C ) ) ) ^ 2 )
6310recni 8845 . . . . . . . . . . 11  |-  ( normh `  ( B  -h  C
) )  e.  CC
6416, 63sqmuli 11182 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( B  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
65 sq2 11194 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
6665oveq1i 5830 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
6762, 64, 663eqtri 2309 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
682, 8hvsubcan2i 21636 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  +h  ( A  -h  B ) )  =  ( 2  .h  A
)
6968oveq1i 5830 . . . . . . . . . . . . . . 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 21626 . . . . . . . . . . . . . . 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 21587 . . . . . . . . . . . . . . . 16  |-  ( 2  .h  A )  e. 
~H
7271, 17hvsubvali 21593 . . . . . . . . . . . . . . 15  |-  ( ( 2  .h  A )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  A )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
7369, 70, 723eqtr4i 2315 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( 2  .h  C ) ) )  +h  ( A  -h  B ) )  =  ( ( 2  .h  A )  -h  ( 2  .h  C
) )
7438oveq1i 5830 . . . . . . . . . . . . . 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 21624 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( A  -h  C ) )  =  ( ( 2  .h  A )  -h  (
2  .h  C ) )
7673, 74, 753eqtr4i 2315 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( A  -h  B ) )  =  ( 2  .h  ( A  -h  C ) )
7776fveq2i 5489 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( A  -h  C ) ) )
7816, 4norm-iii-i 21711 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( A  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )
7959oveq1i 5830 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8077, 78, 793eqtri 2309 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8180oveq1i 5830 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( A  -h  C ) ) ) ^ 2 )
825recni 8845 . . . . . . . . . . 11  |-  ( normh `  ( A  -h  C
) )  e.  CC
8316, 82sqmuli 11182 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( A  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8465oveq1i 5830 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8581, 83, 843eqtri 2309 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8667, 85oveq12i 5832 . . . . . . . 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 2308 . . . . . . 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 21726 . . . . . . 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 2305 . . . . . 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 5830 . . . . 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 8838 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  e.  CC
9216, 25mulcli 8838 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( A  -h  B
) ) ^ 2 ) )  e.  CC
9391, 92, 16, 31divdiri 9513 . . . . 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 9502 . . . . . 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 9502 . . . . . 6  |-  ( ( 2  x.  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  / 
2 )  =  ( ( normh `  ( A  -h  B ) ) ^
2 )
9694, 95oveq12i 5832 . . . . 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 2309 . . . 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 9514 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
99 4d2e2 9872 . . . . . . 7  |-  ( 4  /  2 )  =  2
10099oveq1i 5830 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10198, 100eqtri 2305 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10226, 29, 16, 31div23i 9514 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
10399oveq1i 5830 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
104102, 103eqtri 2305 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
105101, 104oveq12i 5832 . . . 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 2313 . . 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 9131 . 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 2289 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 1624    e. wcel 1685   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   0cc0 8733   1c1 8734    + caddc 8736    x. cmul 8738    <_ cle 8864    - cmin 9033   -ucneg 9034    / cdiv 9419   2c2 9791   4c4 9793   ^cexp 11099   abscabs 11714   ~Hchil 21492    +h cva 21493    .h csm 21494   normhcno 21496    -h cmv 21498
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-hfvadd 21573  ax-hvcom 21574  ax-hvass 21575  ax-hv0cl 21576  ax-hvaddid 21577  ax-hfvmul 21578  ax-hvmulid 21579  ax-hvmulass 21580  ax-hvdistr1 21581  ax-hvdistr2 21582  ax-hvmul0 21583  ax-hfi 21651  ax-his1 21654  ax-his2 21655  ax-his3 21656  ax-his4 21657
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  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 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11042  df-exp 11100  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-hnorm 21541  df-hvsub 21544
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