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Theorem normpar2i 21565
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 9695 . . . . . 6  |-  2  e.  RR
2 normpar2.1 . . . . . . . . 9  |-  A  e. 
~H
3 normpar2.3 . . . . . . . . 9  |-  C  e. 
~H
42, 3hvsubcli 21431 . . . . . . . 8  |-  ( A  -h  C )  e. 
~H
54normcli 21540 . . . . . . 7  |-  ( normh `  ( A  -h  C
) )  e.  RR
65resqcli 11067 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  RR
71, 6remulcli 8731 . . . . 5  |-  ( 2  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  RR
8 normpar2.2 . . . . . . . . 9  |-  B  e. 
~H
98, 3hvsubcli 21431 . . . . . . . 8  |-  ( B  -h  C )  e. 
~H
109normcli 21540 . . . . . . 7  |-  ( normh `  ( B  -h  C
) )  e.  RR
1110resqcli 11067 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  RR
121, 11remulcli 8731 . . . . 5  |-  ( 2  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  RR
137, 12readdcli 8730 . . . 4  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  RR
1413recni 8729 . . 3  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  CC
152, 8hvaddcli 21428 . . . . . . 7  |-  ( A  +h  B )  e. 
~H
16 2cn 9696 . . . . . . . 8  |-  2  e.  CC
1716, 3hvmulcli 21424 . . . . . . 7  |-  ( 2  .h  C )  e. 
~H
1815, 17hvsubcli 21431 . . . . . 6  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  e. 
~H
1918normcli 21540 . . . . 5  |-  ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) )  e.  RR
2019resqcli 11067 . . . 4  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  RR
2120recni 8729 . . 3  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  CC
222, 8hvsubcli 21431 . . . . . 6  |-  ( A  -h  B )  e. 
~H
2322normcli 21540 . . . . 5  |-  ( normh `  ( A  -h  B
) )  e.  RR
2423resqcli 11067 . . . 4  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  RR
2524recni 8729 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  CC
26 4cn 9700 . . . . . 6  |-  4  e.  CC
276recni 8729 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  CC
2826, 27mulcli 8722 . . . . 5  |-  ( 4  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  CC
2911recni 8729 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  CC
3026, 29mulcli 8722 . . . . 5  |-  ( 4  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  CC
31 2ne0 9709 . . . . 5  |-  2  =/=  0
3228, 30, 16, 31divdiri 9397 . . . 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 8883 . . . . . . . 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 9693 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
3534, 17hvmulcli 21424 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( 2  .h  C ) )  e.  ~H
3634, 22hvmulcli 21424 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( A  -h  B ) )  e.  ~H
3715, 35, 36hvadd32i 21463 . . . . . . . . . . . . . . 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 21430 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  =  ( ( A  +h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
3938oveq1i 5720 . . . . . . . . . . . . . . 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 21424 . . . . . . . . . . . . . . . . 17  |-  ( 2  .h  B )  e. 
~H
4140, 17hvsubvali 21430 . . . . . . . . . . . . . . . 16  |-  ( ( 2  .h  B )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
422, 8hvcomi 21429 . . . . . . . . . . . . . . . . . . 19  |-  ( A  +h  B )  =  ( B  +h  A
)
432, 8hvnegdii 21471 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1  .h  ( A  -h  B ) )  =  ( B  -h  A )
4442, 43oveq12i 5722 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( B  +h  A )  +h  ( B  -h  A ) )
458, 2hvsubcan2i 21473 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  +h  A )  +h  ( B  -h  A ) )  =  ( 2  .h  B
)
4644, 45eqtri 2273 . . . . . . . . . . . . . . . . 17  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( 2  .h  B
)
4746oveq1i 5720 . . . . . . . . . . . . . . . 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 2276 . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5018, 22hvsubvali 21430 . . . . . . . . . . . . . 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 21461 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( B  -h  C ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5249, 50, 513eqtr4i 2283 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  -h  ( A  -h  B ) )  =  ( 2  .h  ( B  -h  C ) )
5352fveq2i 5380 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( B  -h  C ) ) )
5416, 9norm-iii-i 21548 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( B  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )
55 0re 8718 . . . . . . . . . . . . . . 15  |-  0  e.  RR
56 2pos 9708 . . . . . . . . . . . . . . 15  |-  0  <  2
5755, 1, 56ltleii 8821 . . . . . . . . . . . . . 14  |-  0  <_  2
581absidi 11738 . . . . . . . . . . . . . 14  |-  ( 0  <_  2  ->  ( abs `  2 )  =  2 )
5957, 58ax-mp 10 . . . . . . . . . . . . 13  |-  ( abs `  2 )  =  2
6059oveq1i 5720 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6153, 54, 603eqtri 2277 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6261oveq1i 5720 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( B  -h  C ) ) ) ^ 2 )
6310recni 8729 . . . . . . . . . . 11  |-  ( normh `  ( B  -h  C
) )  e.  CC
6416, 63sqmuli 11065 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( B  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
65 sq2 11077 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
6665oveq1i 5720 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
6762, 64, 663eqtri 2277 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
682, 8hvsubcan2i 21473 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  +h  ( A  -h  B ) )  =  ( 2  .h  A
)
6968oveq1i 5720 . . . . . . . . . . . . . . 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 21463 . . . . . . . . . . . . . . 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 21424 . . . . . . . . . . . . . . . 16  |-  ( 2  .h  A )  e. 
~H
7271, 17hvsubvali 21430 . . . . . . . . . . . . . . 15  |-  ( ( 2  .h  A )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  A )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
7369, 70, 723eqtr4i 2283 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( 2  .h  C ) ) )  +h  ( A  -h  B ) )  =  ( ( 2  .h  A )  -h  ( 2  .h  C
) )
7438oveq1i 5720 . . . . . . . . . . . . . 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 21461 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( A  -h  C ) )  =  ( ( 2  .h  A )  -h  (
2  .h  C ) )
7673, 74, 753eqtr4i 2283 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( A  -h  B ) )  =  ( 2  .h  ( A  -h  C ) )
7776fveq2i 5380 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( A  -h  C ) ) )
7816, 4norm-iii-i 21548 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( A  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )
7959oveq1i 5720 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8077, 78, 793eqtri 2277 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8180oveq1i 5720 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( A  -h  C ) ) ) ^ 2 )
825recni 8729 . . . . . . . . . . 11  |-  ( normh `  ( A  -h  C
) )  e.  CC
8316, 82sqmuli 11065 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( A  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8465oveq1i 5720 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8581, 83, 843eqtri 2277 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8667, 85oveq12i 5722 . . . . . . . 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 2276 . . . . . . 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 21563 . . . . . . 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 2273 . . . . . 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 5720 . . . . 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 8722 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  e.  CC
9216, 25mulcli 8722 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( A  -h  B
) ) ^ 2 ) )  e.  CC
9391, 92, 16, 31divdiri 9397 . . . . 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 9386 . . . . . 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 9386 . . . . . 6  |-  ( ( 2  x.  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  / 
2 )  =  ( ( normh `  ( A  -h  B ) ) ^
2 )
9694, 95oveq12i 5722 . . . . 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 2277 . . . 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 9398 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
99 4d2e2 9755 . . . . . . 7  |-  ( 4  /  2 )  =  2
10099oveq1i 5720 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10198, 100eqtri 2273 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10226, 29, 16, 31div23i 9398 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
10399oveq1i 5720 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
104102, 103eqtri 2273 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
105101, 104oveq12i 5722 . . . 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 2281 . . 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 9015 . 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 2257 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 1619    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    <_ cle 8748    - cmin 8917   -ucneg 8918    / cdiv 9303   2c2 9675   4c4 9677   ^cexp 10982   abscabs 11596   ~Hchil 21329    +h cva 21330    .h csm 21331   normhcno 21333    -h cmv 21335
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-hfvadd 21410  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvmulass 21417  ax-hvdistr1 21418  ax-hvdistr2 21419  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his2 21492  ax-his3 21493  ax-his4 21494
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-hnorm 21378  df-hvsub 21381
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