HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  normpar2i Unicode version

Theorem normpar2i 22611
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 10025 . . . . . 6  |-  2  e.  RR
2 normpar2.1 . . . . . . . . 9  |-  A  e. 
~H
3 normpar2.3 . . . . . . . . 9  |-  C  e. 
~H
42, 3hvsubcli 22477 . . . . . . . 8  |-  ( A  -h  C )  e. 
~H
54normcli 22586 . . . . . . 7  |-  ( normh `  ( A  -h  C
) )  e.  RR
65resqcli 11422 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  RR
71, 6remulcli 9060 . . . . 5  |-  ( 2  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  RR
8 normpar2.2 . . . . . . . . 9  |-  B  e. 
~H
98, 3hvsubcli 22477 . . . . . . . 8  |-  ( B  -h  C )  e. 
~H
109normcli 22586 . . . . . . 7  |-  ( normh `  ( B  -h  C
) )  e.  RR
1110resqcli 11422 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  RR
121, 11remulcli 9060 . . . . 5  |-  ( 2  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  RR
137, 12readdcli 9059 . . . 4  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  RR
1413recni 9058 . . 3  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  CC
152, 8hvaddcli 22474 . . . . . . 7  |-  ( A  +h  B )  e. 
~H
16 2cn 10026 . . . . . . . 8  |-  2  e.  CC
1716, 3hvmulcli 22470 . . . . . . 7  |-  ( 2  .h  C )  e. 
~H
1815, 17hvsubcli 22477 . . . . . 6  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  e. 
~H
1918normcli 22586 . . . . 5  |-  ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) )  e.  RR
2019resqcli 11422 . . . 4  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  RR
2120recni 9058 . . 3  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  CC
222, 8hvsubcli 22477 . . . . . 6  |-  ( A  -h  B )  e. 
~H
2322normcli 22586 . . . . 5  |-  ( normh `  ( A  -h  B
) )  e.  RR
2423resqcli 11422 . . . 4  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  RR
2524recni 9058 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  CC
26 4cn 10030 . . . . . 6  |-  4  e.  CC
276recni 9058 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  CC
2826, 27mulcli 9051 . . . . 5  |-  ( 4  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  CC
2911recni 9058 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  CC
3026, 29mulcli 9051 . . . . 5  |-  ( 4  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  CC
31 2ne0 10039 . . . . 5  |-  2  =/=  0
3228, 30, 16, 31divdiri 9727 . . . 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 9213 . . . . . . . 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 10023 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
3534, 17hvmulcli 22470 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( 2  .h  C ) )  e.  ~H
3634, 22hvmulcli 22470 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( A  -h  B ) )  e.  ~H
3715, 35, 36hvadd32i 22509 . . . . . . . . . . . . . . 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 22476 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  =  ( ( A  +h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
3938oveq1i 6050 . . . . . . . . . . . . . . 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 22470 . . . . . . . . . . . . . . . . 17  |-  ( 2  .h  B )  e. 
~H
4140, 17hvsubvali 22476 . . . . . . . . . . . . . . . 16  |-  ( ( 2  .h  B )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
422, 8hvcomi 22475 . . . . . . . . . . . . . . . . . . 19  |-  ( A  +h  B )  =  ( B  +h  A
)
432, 8hvnegdii 22517 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1  .h  ( A  -h  B ) )  =  ( B  -h  A )
4442, 43oveq12i 6052 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( B  +h  A )  +h  ( B  -h  A ) )
458, 2hvsubcan2i 22519 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  +h  A )  +h  ( B  -h  A ) )  =  ( 2  .h  B
)
4644, 45eqtri 2424 . . . . . . . . . . . . . . . . 17  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( 2  .h  B
)
4746oveq1i 6050 . . . . . . . . . . . . . . . 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 2427 . . . . . . . . . . . . . . 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 2434 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5018, 22hvsubvali 22476 . . . . . . . . . . . . . 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 22507 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( B  -h  C ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5249, 50, 513eqtr4i 2434 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  -h  ( A  -h  B ) )  =  ( 2  .h  ( B  -h  C ) )
5352fveq2i 5690 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( B  -h  C ) ) )
5416, 9norm-iii-i 22594 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( B  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )
55 0re 9047 . . . . . . . . . . . . . . 15  |-  0  e.  RR
56 2pos 10038 . . . . . . . . . . . . . . 15  |-  0  <  2
5755, 1, 56ltleii 9152 . . . . . . . . . . . . . 14  |-  0  <_  2
581absidi 12136 . . . . . . . . . . . . . 14  |-  ( 0  <_  2  ->  ( abs `  2 )  =  2 )
5957, 58ax-mp 8 . . . . . . . . . . . . 13  |-  ( abs `  2 )  =  2
6059oveq1i 6050 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6153, 54, 603eqtri 2428 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6261oveq1i 6050 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( B  -h  C ) ) ) ^ 2 )
6310recni 9058 . . . . . . . . . . 11  |-  ( normh `  ( B  -h  C
) )  e.  CC
6416, 63sqmuli 11420 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( B  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
65 sq2 11432 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
6665oveq1i 6050 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
6762, 64, 663eqtri 2428 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
682, 8hvsubcan2i 22519 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  +h  ( A  -h  B ) )  =  ( 2  .h  A
)
6968oveq1i 6050 . . . . . . . . . . . . . . 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 22509 . . . . . . . . . . . . . . 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 22470 . . . . . . . . . . . . . . . 16  |-  ( 2  .h  A )  e. 
~H
7271, 17hvsubvali 22476 . . . . . . . . . . . . . . 15  |-  ( ( 2  .h  A )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  A )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
7369, 70, 723eqtr4i 2434 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( 2  .h  C ) ) )  +h  ( A  -h  B ) )  =  ( ( 2  .h  A )  -h  ( 2  .h  C
) )
7438oveq1i 6050 . . . . . . . . . . . . . 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 22507 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( A  -h  C ) )  =  ( ( 2  .h  A )  -h  (
2  .h  C ) )
7673, 74, 753eqtr4i 2434 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( A  -h  B ) )  =  ( 2  .h  ( A  -h  C ) )
7776fveq2i 5690 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( A  -h  C ) ) )
7816, 4norm-iii-i 22594 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( A  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )
7959oveq1i 6050 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8077, 78, 793eqtri 2428 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8180oveq1i 6050 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( A  -h  C ) ) ) ^ 2 )
825recni 9058 . . . . . . . . . . 11  |-  ( normh `  ( A  -h  C
) )  e.  CC
8316, 82sqmuli 11420 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( A  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8465oveq1i 6050 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8581, 83, 843eqtri 2428 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8667, 85oveq12i 6052 . . . . . . . 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 2427 . . . . . . 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 22609 . . . . . . 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 2424 . . . . . 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 6050 . . . . 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 9051 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  e.  CC
9216, 25mulcli 9051 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( A  -h  B
) ) ^ 2 ) )  e.  CC
9391, 92, 16, 31divdiri 9727 . . . . 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 9716 . . . . . 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 9716 . . . . . 6  |-  ( ( 2  x.  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  / 
2 )  =  ( ( normh `  ( A  -h  B ) ) ^
2 )
9694, 95oveq12i 6052 . . . . 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 2428 . . . 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 9728 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
99 4d2e2 10088 . . . . . . 7  |-  ( 4  /  2 )  =  2
10099oveq1i 6050 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10198, 100eqtri 2424 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10226, 29, 16, 31div23i 9728 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
10399oveq1i 6050 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
104102, 103eqtri 2424 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
105101, 104oveq12i 6052 . . . 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 2432 . . 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 9345 . 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 2408 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 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   2c2 10005   4c4 10007   ^cexp 11337   abscabs 11994   ~Hchil 22375    +h cva 22376    .h csm 22377   normhcno 22379    -h cmv 22381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-hnorm 22424  df-hvsub 22427
  Copyright terms: Public domain W3C validator