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Theorem ipidsq 21232
Description: The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ipid.1  |-  X  =  ( BaseSet `  U )
ipid.6  |-  N  =  ( normCV `  U )
ipid.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
ipidsq  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P A )  =  ( ( N `  A ) ^ 2 ) )

Proof of Theorem ipidsq
StepHypRef Expression
1 ipid.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 eqid 2256 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2256 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 ipid.6 . . . 4  |-  N  =  ( normCV `  U )
5 ipid.7 . . . 4  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval2 21226 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  e.  X )  ->  ( A P A )  =  ( ( ( ( ( N `  ( A ( +v `  U ) A ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A ( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
763anidm23 1246 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P A )  =  ( ( ( ( ( N `  ( A ( +v `  U ) A ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A ( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
81, 2, 3nv2 21136 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) A )  =  ( 2 ( .s OLD `  U
) A ) )
98fveq2d 5448 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( +v `  U
) A ) )  =  ( N `  ( 2 ( .s
OLD `  U ) A ) ) )
10 2re 9769 . . . . . . . . . . . 12  |-  2  e.  RR
11 0re 8792 . . . . . . . . . . . . 13  |-  0  e.  RR
12 2pos 9782 . . . . . . . . . . . . 13  |-  0  <  2
1311, 10, 12ltleii 8895 . . . . . . . . . . . 12  |-  0  <_  2
1410, 13pm3.2i 443 . . . . . . . . . . 11  |-  ( 2  e.  RR  /\  0  <_  2 )
151, 3, 4nvsge0 21175 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  (
2  e.  RR  /\  0  <_  2 )  /\  A  e.  X )  ->  ( N `  (
2 ( .s OLD `  U ) A ) )  =  ( 2  x.  ( N `  A ) ) )
1614, 15mp3an2 1270 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 2
( .s OLD `  U
) A ) )  =  ( 2  x.  ( N `  A
) ) )
179, 16eqtrd 2288 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( +v `  U
) A ) )  =  ( 2  x.  ( N `  A
) ) )
1817oveq1d 5793 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  ( A ( +v `  U ) A ) ) ^ 2 )  =  ( ( 2  x.  ( N `  A ) ) ^
2 ) )
191, 4nvcl 21171 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
2019recnd 8815 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
21 2cn 9770 . . . . . . . . . . 11  |-  2  e.  CC
22 2nn0 9935 . . . . . . . . . . 11  |-  2  e.  NN0
23 mulexp 11093 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  ( N `  A )  e.  CC  /\  2  e.  NN0 )  ->  (
( 2  x.  ( N `  A )
) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( ( N `  A ) ^ 2 ) ) )
2421, 22, 23mp3an13 1273 . . . . . . . . . 10  |-  ( ( N `  A )  e.  CC  ->  (
( 2  x.  ( N `  A )
) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( ( N `  A ) ^ 2 ) ) )
2520, 24syl 17 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 2  x.  ( N `  A )
) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( ( N `  A ) ^ 2 ) ) )
26 sq2 11151 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
2726oveq1i 5788 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  x.  ( ( N `
 A ) ^
2 ) )  =  ( 4  x.  (
( N `  A
) ^ 2 ) )
2825, 27syl6eq 2304 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 2  x.  ( N `  A )
) ^ 2 )  =  ( 4  x.  ( ( N `  A ) ^ 2 ) ) )
2918, 28eqtrd 2288 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  ( A ( +v `  U ) A ) ) ^ 2 )  =  ( 4  x.  ( ( N `  A ) ^ 2 ) ) )
30 eqid 2256 . . . . . . . . . . . 12  |-  ( 0vec `  U )  =  (
0vec `  U )
311, 2, 3, 30nvrinv 21157 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) )  =  ( 0vec `  U ) )
3231fveq2d 5448 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) A ) ) )  =  ( N `
 ( 0vec `  U
) ) )
3330, 4nvz0 21180 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( N `  ( 0vec `  U )
)  =  0 )
3433adantr 453 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  0 )
3532, 34eqtrd 2288 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) A ) ) )  =  0 )
3635oveq1d 5793 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 )  =  ( 0 ^ 2 ) )
37 sq0 11147 . . . . . . . 8  |-  ( 0 ^ 2 )  =  0
3836, 37syl6eq 2304 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 )  =  0 )
3929, 38oveq12d 5796 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( N `  ( A ( +v `  U ) A ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) A ) ) ) ^ 2 ) )  =  ( ( 4  x.  ( ( N `  A ) ^ 2 ) )  -  0 ) )
40 4cn 9774 . . . . . . . 8  |-  4  e.  CC
4120sqcld 11195 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
) ^ 2 )  e.  CC )
42 mulcl 8775 . . . . . . . 8  |-  ( ( 4  e.  CC  /\  ( ( N `  A ) ^ 2 )  e.  CC )  ->  ( 4  x.  ( ( N `  A ) ^ 2 ) )  e.  CC )
4340, 41, 42sylancr 647 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
4  x.  ( ( N `  A ) ^ 2 ) )  e.  CC )
4443subid1d 9100 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 4  x.  (
( N `  A
) ^ 2 ) )  -  0 )  =  ( 4  x.  ( ( N `  A ) ^ 2 ) ) )
4539, 44eqtrd 2288 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( N `  ( A ( +v `  U ) A ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) A ) ) ) ^ 2 ) )  =  ( 4  x.  ( ( N `
 A ) ^
2 ) ) )
46 1re 8791 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
4746renegcli 9062 . . . . . . . . . . . . . . . 16  |-  -u 1  e.  RR
48 absreim 11729 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  -u 1  e.  RR )  ->  ( abs `  (
1  +  ( _i  x.  -u 1 ) ) )  =  ( sqr `  ( ( 1 ^ 2 )  +  (
-u 1 ^ 2 ) ) ) )
4946, 47, 48mp2an 656 . . . . . . . . . . . . . . 15  |-  ( abs `  ( 1  +  ( _i  x.  -u 1
) ) )  =  ( sqr `  (
( 1 ^ 2 )  +  ( -u
1 ^ 2 ) ) )
50 ax-icn 8750 . . . . . . . . . . . . . . . . . . 19  |-  _i  e.  CC
51 ax-1cn 8749 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
5250, 51mulneg2i 9180 . . . . . . . . . . . . . . . . . 18  |-  ( _i  x.  -u 1 )  = 
-u ( _i  x.  1 )
5350mulid1i 8793 . . . . . . . . . . . . . . . . . . 19  |-  ( _i  x.  1 )  =  _i
5453negeqi 8999 . . . . . . . . . . . . . . . . . 18  |-  -u (
_i  x.  1 )  =  -u _i
5552, 54eqtri 2276 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  -u 1 )  = 
-u _i
5655oveq2i 5789 . . . . . . . . . . . . . . . 16  |-  ( 1  +  ( _i  x.  -u 1 ) )  =  ( 1  +  -u _i )
5756fveq2i 5447 . . . . . . . . . . . . . . 15  |-  ( abs `  ( 1  +  ( _i  x.  -u 1
) ) )  =  ( abs `  (
1  +  -u _i ) )
58 sqneg 11116 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  CC  ->  ( -u 1 ^ 2 )  =  ( 1 ^ 2 ) )
5951, 58ax-mp 10 . . . . . . . . . . . . . . . . 17  |-  ( -u
1 ^ 2 )  =  ( 1 ^ 2 )
6059oveq2i 5789 . . . . . . . . . . . . . . . 16  |-  ( ( 1 ^ 2 )  +  ( -u 1 ^ 2 ) )  =  ( ( 1 ^ 2 )  +  ( 1 ^ 2 ) )
6160fveq2i 5447 . . . . . . . . . . . . . . 15  |-  ( sqr `  ( ( 1 ^ 2 )  +  (
-u 1 ^ 2 ) ) )  =  ( sqr `  (
( 1 ^ 2 )  +  ( 1 ^ 2 ) ) )
6249, 57, 613eqtr3i 2284 . . . . . . . . . . . . . 14  |-  ( abs `  ( 1  +  -u _i ) )  =  ( sqr `  ( ( 1 ^ 2 )  +  ( 1 ^ 2 ) ) )
63 absreim 11729 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR  /\  1  e.  RR )  ->  ( abs `  (
1  +  ( _i  x.  1 ) ) )  =  ( sqr `  ( ( 1 ^ 2 )  +  ( 1 ^ 2 ) ) ) )
6446, 46, 63mp2an 656 . . . . . . . . . . . . . 14  |-  ( abs `  ( 1  +  ( _i  x.  1 ) ) )  =  ( sqr `  ( ( 1 ^ 2 )  +  ( 1 ^ 2 ) ) )
6553oveq2i 5789 . . . . . . . . . . . . . . 15  |-  ( 1  +  ( _i  x.  1 ) )  =  ( 1  +  _i )
6665fveq2i 5447 . . . . . . . . . . . . . 14  |-  ( abs `  ( 1  +  ( _i  x.  1 ) ) )  =  ( abs `  ( 1  +  _i ) )
6762, 64, 663eqtr2i 2282 . . . . . . . . . . . . 13  |-  ( abs `  ( 1  +  -u _i ) )  =  ( abs `  ( 1  +  _i ) )
6867oveq1i 5788 . . . . . . . . . . . 12  |-  ( ( abs `  ( 1  +  -u _i ) )  x.  ( N `  A ) )  =  ( ( abs `  (
1  +  _i ) )  x.  ( N `
 A ) )
6950negcli 9068 . . . . . . . . . . . . . 14  |-  -u _i  e.  CC
7051, 69addcli 8795 . . . . . . . . . . . . 13  |-  ( 1  +  -u _i )  e.  CC
711, 3, 4nvs 21174 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
1  +  -u _i )  e.  CC  /\  A  e.  X )  ->  ( N `  ( (
1  +  -u _i ) ( .s OLD `  U ) A ) )  =  ( ( abs `  ( 1  +  -u _i ) )  x.  ( N `  A ) ) )
7270, 71mp3an2 1270 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( (
1  +  -u _i ) ( .s OLD `  U ) A ) )  =  ( ( abs `  ( 1  +  -u _i ) )  x.  ( N `  A ) ) )
7351, 50addcli 8795 . . . . . . . . . . . . 13  |-  ( 1  +  _i )  e.  CC
741, 3, 4nvs 21174 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
1  +  _i )  e.  CC  /\  A  e.  X )  ->  ( N `  ( (
1  +  _i ) ( .s OLD `  U
) A ) )  =  ( ( abs `  ( 1  +  _i ) )  x.  ( N `  A )
) )
7573, 74mp3an2 1270 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( (
1  +  _i ) ( .s OLD `  U
) A ) )  =  ( ( abs `  ( 1  +  _i ) )  x.  ( N `  A )
) )
7668, 72, 753eqtr4a 2314 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( (
1  +  -u _i ) ( .s OLD `  U ) A ) )  =  ( N `
 ( ( 1  +  _i ) ( .s OLD `  U
) A ) ) )
771, 2, 3nvdir 21135 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  NrmCVec  /\  (
1  e.  CC  /\  -u _i  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  -u _i ) ( .s OLD `  U ) A )  =  ( ( 1 ( .s OLD `  U
) A ) ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) )
7851, 77mp3anr1 1279 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  ( -u _i  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  -u _i ) ( .s OLD `  U ) A )  =  ( ( 1 ( .s OLD `  U
) A ) ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) )
7969, 78mpanr1 667 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1  +  -u _i ) ( .s OLD `  U ) A )  =  ( ( 1 ( .s OLD `  U
) A ) ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) )
801, 3nvsid 21131 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .s OLD `  U ) A )  =  A )
8180oveq1d 5793 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u _i ( .s
OLD `  U ) A ) )  =  ( A ( +v
`  U ) (
-u _i ( .s
OLD `  U ) A ) ) )
8279, 81eqtrd 2288 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1  +  -u _i ) ( .s OLD `  U ) A )  =  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) )
8382fveq2d 5448 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( (
1  +  -u _i ) ( .s OLD `  U ) A ) )  =  ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) )
841, 2, 3nvdir 21135 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  NrmCVec  /\  (
1  e.  CC  /\  _i  e.  CC  /\  A  e.  X ) )  -> 
( ( 1  +  _i ) ( .s
OLD `  U ) A )  =  ( ( 1 ( .s
OLD `  U ) A ) ( +v
`  U ) ( _i ( .s OLD `  U ) A ) ) )
8551, 84mp3anr1 1279 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  (
_i  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  _i ) ( .s OLD `  U
) A )  =  ( ( 1 ( .s OLD `  U
) A ) ( +v `  U ) ( _i ( .s
OLD `  U ) A ) ) )
8650, 85mpanr1 667 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1  +  _i ) ( .s OLD `  U ) A )  =  ( ( 1 ( .s OLD `  U
) A ) ( +v `  U ) ( _i ( .s
OLD `  U ) A ) ) )
8780oveq1d 5793 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1 ( .s
OLD `  U ) A ) ( +v
`  U ) ( _i ( .s OLD `  U ) A ) )  =  ( A ( +v `  U
) ( _i ( .s OLD `  U
) A ) ) )
8886, 87eqtrd 2288 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1  +  _i ) ( .s OLD `  U ) A )  =  ( A ( +v `  U ) ( _i ( .s
OLD `  U ) A ) ) )
8988fveq2d 5448 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( (
1  +  _i ) ( .s OLD `  U
) A ) )  =  ( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) )
9076, 83, 893eqtr3d 2296 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) A ) ) )  =  ( N `
 ( A ( +v `  U ) ( _i ( .s
OLD `  U ) A ) ) ) )
9190oveq1d 5793 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 )  =  ( ( N `
 ( A ( +v `  U ) ( _i ( .s
OLD `  U ) A ) ) ) ^ 2 ) )
9291oveq2d 5794 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) )  =  ( ( ( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( _i ( .s
OLD `  U ) A ) ) ) ^ 2 ) ) )
931, 2, 3, 4, 5ipval2lem4 21225 . . . . . . . . . . 11  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  e.  X )  /\  _i  e.  CC )  ->  ( ( N `
 ( A ( +v `  U ) ( _i ( .s
OLD `  U ) A ) ) ) ^ 2 )  e.  CC )
9450, 93mpan2 655 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  e.  X )  ->  (
( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  e.  CC )
95943anidm23 1246 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  e.  CC )
9695subidd 9099 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( _i ( .s
OLD `  U ) A ) ) ) ^ 2 ) )  =  0 )
9792, 96eqtrd 2288 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) )  =  0 )
9897oveq2d 5794 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
_i  x.  ( (
( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) )  =  ( _i  x.  0 ) )
9950mul01i 8956 . . . . . 6  |-  ( _i  x.  0 )  =  0
10098, 99syl6eq 2304 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
_i  x.  ( (
( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) )  =  0 )
10145, 100oveq12d 5796 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( N `
 ( A ( +v `  U ) A ) ) ^
2 )  -  (
( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  =  ( ( 4  x.  ( ( N `  A ) ^ 2 ) )  +  0 ) )
10243addid1d 8966 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 4  x.  (
( N `  A
) ^ 2 ) )  +  0 )  =  ( 4  x.  ( ( N `  A ) ^ 2 ) ) )
103101, 102eqtr2d 2289 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
4  x.  ( ( N `  A ) ^ 2 ) )  =  ( ( ( ( N `  ( A ( +v `  U ) A ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A ( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) ) )
104103oveq1d 5793 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 4  x.  (
( N `  A
) ^ 2 ) )  /  4 )  =  ( ( ( ( ( N `  ( A ( +v `  U ) A ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A ( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( N `
 ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
105 4re 9773 . . . . 5  |-  4  e.  RR
106 4pos 9786 . . . . 5  |-  0  <  4
107105, 106gt0ne0ii 9263 . . . 4  |-  4  =/=  0
108 divcan3 9402 . . . 4  |-  ( ( ( ( N `  A ) ^ 2 )  e.  CC  /\  4  e.  CC  /\  4  =/=  0 )  ->  (
( 4  x.  (
( N `  A
) ^ 2 ) )  /  4 )  =  ( ( N `
 A ) ^
2 ) )
10940, 107, 108mp3an23 1274 . . 3  |-  ( ( ( N `  A
) ^ 2 )  e.  CC  ->  (
( 4  x.  (
( N `  A
) ^ 2 ) )  /  4 )  =  ( ( N `
 A ) ^
2 ) )
11041, 109syl 17 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 4  x.  (
( N `  A
) ^ 2 ) )  /  4 )  =  ( ( N `
 A ) ^
2 ) )
1117, 104, 1103eqtr2d 2294 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P A )  =  ( ( N `  A ) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692   _ici 8693    + caddc 8694    x. cmul 8696    <_ cle 8822    - cmin 8991   -ucneg 8992    / cdiv 9377   2c2 9749   4c4 9751   NN0cn0 9918   ^cexp 11056   sqrcsqr 11669   abscabs 11670   NrmCVeccnv 21086   +vcpv 21087   BaseSetcba 21088   .s
OLDcns 21089   0veccn0v 21090   normCVcnmcv 21092   .i OLDcdip 21219
This theorem is referenced by:  ipnm  21233  ipz  21241  pythi  21374  siilem1  21375  hlipgt0  21439  htthlem  21443
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-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  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
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-int 3823  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-se 4311  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-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-oi 7179  df-card 7526  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-fz 10735  df-fzo 10823  df-seq 10999  df-exp 11057  df-hash 11290  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-clim 11913  df-sum 12110  df-grpo 20804  df-gid 20805  df-ginv 20806  df-ablo 20895  df-vc 21048  df-nv 21094  df-va 21097  df-ba 21098  df-sm 21099  df-0v 21100  df-nmcv 21102  df-dip 21220
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