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Theorem normpar2i 21681
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 9769 . . . . . 6  |-  2  e.  RR
2 normpar2.1 . . . . . . . . 9  |-  A  e. 
~H
3 normpar2.3 . . . . . . . . 9  |-  C  e. 
~H
42, 3hvsubcli 21547 . . . . . . . 8  |-  ( A  -h  C )  e. 
~H
54normcli 21656 . . . . . . 7  |-  ( normh `  ( A  -h  C
) )  e.  RR
65resqcli 11141 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  RR
71, 6remulcli 8805 . . . . 5  |-  ( 2  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  RR
8 normpar2.2 . . . . . . . . 9  |-  B  e. 
~H
98, 3hvsubcli 21547 . . . . . . . 8  |-  ( B  -h  C )  e. 
~H
109normcli 21656 . . . . . . 7  |-  ( normh `  ( B  -h  C
) )  e.  RR
1110resqcli 11141 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  RR
121, 11remulcli 8805 . . . . 5  |-  ( 2  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  RR
137, 12readdcli 8804 . . . 4  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  RR
1413recni 8803 . . 3  |-  ( ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  +  ( 2  x.  (
( normh `  ( B  -h  C ) ) ^
2 ) ) )  e.  CC
152, 8hvaddcli 21544 . . . . . . 7  |-  ( A  +h  B )  e. 
~H
16 2cn 9770 . . . . . . . 8  |-  2  e.  CC
1716, 3hvmulcli 21540 . . . . . . 7  |-  ( 2  .h  C )  e. 
~H
1815, 17hvsubcli 21547 . . . . . 6  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  e. 
~H
1918normcli 21656 . . . . 5  |-  ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) )  e.  RR
2019resqcli 11141 . . . 4  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  RR
2120recni 8803 . . 3  |-  ( (
normh `  ( ( A  +h  B )  -h  ( 2  .h  C
) ) ) ^
2 )  e.  CC
222, 8hvsubcli 21547 . . . . . 6  |-  ( A  -h  B )  e. 
~H
2322normcli 21656 . . . . 5  |-  ( normh `  ( A  -h  B
) )  e.  RR
2423resqcli 11141 . . . 4  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  RR
2524recni 8803 . . 3  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  e.  CC
26 4cn 9774 . . . . . 6  |-  4  e.  CC
276recni 8803 . . . . . 6  |-  ( (
normh `  ( A  -h  C ) ) ^
2 )  e.  CC
2826, 27mulcli 8796 . . . . 5  |-  ( 4  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  e.  CC
2911recni 8803 . . . . . 6  |-  ( (
normh `  ( B  -h  C ) ) ^
2 )  e.  CC
3026, 29mulcli 8796 . . . . 5  |-  ( 4  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  e.  CC
31 2ne0 9783 . . . . 5  |-  2  =/=  0
3228, 30, 16, 31divdiri 9471 . . . 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 8957 . . . . . . . 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 9767 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
3534, 17hvmulcli 21540 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( 2  .h  C ) )  e.  ~H
3634, 22hvmulcli 21540 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( A  -h  B ) )  e.  ~H
3715, 35, 36hvadd32i 21579 . . . . . . . . . . . . . . 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 21546 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  -h  ( 2  .h  C ) )  =  ( ( A  +h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
3938oveq1i 5788 . . . . . . . . . . . . . . 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 21540 . . . . . . . . . . . . . . . . 17  |-  ( 2  .h  B )  e. 
~H
4140, 17hvsubvali 21546 . . . . . . . . . . . . . . . 16  |-  ( ( 2  .h  B )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  B )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
422, 8hvcomi 21545 . . . . . . . . . . . . . . . . . . 19  |-  ( A  +h  B )  =  ( B  +h  A
)
432, 8hvnegdii 21587 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1  .h  ( A  -h  B ) )  =  ( B  -h  A )
4442, 43oveq12i 5790 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( B  +h  A )  +h  ( B  -h  A ) )
458, 2hvsubcan2i 21589 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  +h  A )  +h  ( B  -h  A ) )  =  ( 2  .h  B
)
4644, 45eqtri 2276 . . . . . . . . . . . . . . . . 17  |-  ( ( A  +h  B )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( 2  .h  B
)
4746oveq1i 5788 . . . . . . . . . . . . . . . 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 2279 . . . . . . . . . . . . . . 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 2286 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( -u 1  .h  ( A  -h  B
) ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5018, 22hvsubvali 21546 . . . . . . . . . . . . . 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 21577 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( B  -h  C ) )  =  ( ( 2  .h  B )  -h  (
2  .h  C ) )
5249, 50, 513eqtr4i 2286 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  -h  ( A  -h  B ) )  =  ( 2  .h  ( B  -h  C ) )
5352fveq2i 5447 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( B  -h  C ) ) )
5416, 9norm-iii-i 21664 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( B  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )
55 0re 8792 . . . . . . . . . . . . . . 15  |-  0  e.  RR
56 2pos 9782 . . . . . . . . . . . . . . 15  |-  0  <  2
5755, 1, 56ltleii 8895 . . . . . . . . . . . . . 14  |-  0  <_  2
581absidi 11812 . . . . . . . . . . . . . 14  |-  ( 0  <_  2  ->  ( abs `  2 )  =  2 )
5957, 58ax-mp 10 . . . . . . . . . . . . 13  |-  ( abs `  2 )  =  2
6059oveq1i 5788 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( B  -h  C ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6153, 54, 603eqtri 2280 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  -h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( B  -h  C ) ) )
6261oveq1i 5788 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( B  -h  C ) ) ) ^ 2 )
6310recni 8803 . . . . . . . . . . 11  |-  ( normh `  ( B  -h  C
) )  e.  CC
6416, 63sqmuli 11139 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( B  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
65 sq2 11151 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
6665oveq1i 5788 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
6762, 64, 663eqtri 2280 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  -h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
682, 8hvsubcan2i 21589 . . . . . . . . . . . . . . . 16  |-  ( ( A  +h  B )  +h  ( A  -h  B ) )  =  ( 2  .h  A
)
6968oveq1i 5788 . . . . . . . . . . . . . . 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 21579 . . . . . . . . . . . . . . 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 21540 . . . . . . . . . . . . . . . 16  |-  ( 2  .h  A )  e. 
~H
7271, 17hvsubvali 21546 . . . . . . . . . . . . . . 15  |-  ( ( 2  .h  A )  -h  ( 2  .h  C ) )  =  ( ( 2  .h  A )  +h  ( -u 1  .h  ( 2  .h  C ) ) )
7369, 70, 723eqtr4i 2286 . . . . . . . . . . . . . 14  |-  ( ( ( A  +h  B
)  +h  ( -u
1  .h  ( 2  .h  C ) ) )  +h  ( A  -h  B ) )  =  ( ( 2  .h  A )  -h  ( 2  .h  C
) )
7438oveq1i 5788 . . . . . . . . . . . . . 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 21577 . . . . . . . . . . . . . 14  |-  ( 2  .h  ( A  -h  C ) )  =  ( ( 2  .h  A )  -h  (
2  .h  C ) )
7673, 74, 753eqtr4i 2286 . . . . . . . . . . . . 13  |-  ( ( ( A  +h  B
)  -h  ( 2  .h  C ) )  +h  ( A  -h  B ) )  =  ( 2  .h  ( A  -h  C ) )
7776fveq2i 5447 . . . . . . . . . . . 12  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( normh `  ( 2  .h  ( A  -h  C ) ) )
7816, 4norm-iii-i 21664 . . . . . . . . . . . 12  |-  ( normh `  ( 2  .h  ( A  -h  C ) ) )  =  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )
7959oveq1i 5788 . . . . . . . . . . . 12  |-  ( ( abs `  2 )  x.  ( normh `  ( A  -h  C ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8077, 78, 793eqtri 2280 . . . . . . . . . . 11  |-  ( normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C
) )  +h  ( A  -h  B ) ) )  =  ( 2  x.  ( normh `  ( A  -h  C ) ) )
8180oveq1i 5788 . . . . . . . . . 10  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( ( 2  x.  ( normh `  ( A  -h  C ) ) ) ^ 2 )
825recni 8803 . . . . . . . . . . 11  |-  ( normh `  ( A  -h  C
) )  e.  CC
8316, 82sqmuli 11139 . . . . . . . . . 10  |-  ( ( 2  x.  ( normh `  ( A  -h  C
) ) ) ^
2 )  =  ( ( 2 ^ 2 )  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8465oveq1i 5788 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8581, 83, 843eqtri 2280 . . . . . . . . 9  |-  ( (
normh `  ( ( ( A  +h  B )  -h  ( 2  .h  C ) )  +h  ( A  -h  B
) ) ) ^
2 )  =  ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
8667, 85oveq12i 5790 . . . . . . . 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 2279 . . . . . . 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 21679 . . . . . . 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 2276 . . . . . 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 5788 . . . . 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 8796 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( ( A  +h  B )  -h  (
2  .h  C ) ) ) ^ 2 ) )  e.  CC
9216, 25mulcli 8796 . . . . . 6  |-  ( 2  x.  ( ( normh `  ( A  -h  B
) ) ^ 2 ) )  e.  CC
9391, 92, 16, 31divdiri 9471 . . . . 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 9460 . . . . . 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 9460 . . . . . 6  |-  ( ( 2  x.  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  / 
2 )  =  ( ( normh `  ( A  -h  B ) ) ^
2 )
9694, 95oveq12i 5790 . . . . 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 2280 . . . 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 9472 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
99 4d2e2 9829 . . . . . . 7  |-  ( 4  /  2 )  =  2
10099oveq1i 5788 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( A  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10198, 100eqtri 2276 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( A  -h  C ) ) ^
2 ) )
10226, 29, 16, 31div23i 9472 . . . . . 6  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( ( 4  /  2
)  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
10399oveq1i 5788 . . . . . 6  |-  ( ( 4  /  2 )  x.  ( ( normh `  ( B  -h  C
) ) ^ 2 ) )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
104102, 103eqtri 2276 . . . . 5  |-  ( ( 4  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )  / 
2 )  =  ( 2  x.  ( (
normh `  ( B  -h  C ) ) ^
2 ) )
105101, 104oveq12i 5790 . . . 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 2284 . . 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 9089 . 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 2260 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 3983   ` cfv 4659  (class class class)co 5778   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    <_ cle 8822    - cmin 8991   -ucneg 8992    / cdiv 9377   2c2 9749   4c4 9751   ^cexp 11056   abscabs 11670   ~Hchil 21445    +h cva 21446    .h csm 21447   normhcno 21449    -h cmv 21451
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 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-hfvadd 21526  ax-hvcom 21527  ax-hvass 21528  ax-hv0cl 21529  ax-hvaddid 21530  ax-hfvmul 21531  ax-hvmulid 21532  ax-hvmulass 21533  ax-hvdistr1 21534  ax-hvdistr2 21535  ax-hvmul0 21536  ax-hfi 21604  ax-his1 21607  ax-his2 21608  ax-his3 21609  ax-his4 21610
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-sup 7148  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-seq 10999  df-exp 11057  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-hnorm 21494  df-hvsub 21497
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