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Theorem dipcj 21251
Description: The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ipcl.1  |-  X  =  ( BaseSet `  U )
ipcl.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
dipcj  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )

Proof of Theorem dipcj
StepHypRef Expression
1 ipcl.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 eqid 2258 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2258 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2258 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . . 4  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval2 21241 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )
76fveq2d 5462 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( * `  (
( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) ) )
81, 2, 3, 4, 5ipval2 21241 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  ( B P A )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
983com23 1162 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( B P A )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
101, 2, 3, 4, 5ipval2lem3 21239 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  RR )
1110recnd 8829 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  CC )
12 neg1cn 9781 . . . . . . . 8  |-  -u 1  e.  CC
131, 2, 3, 4, 5ipval2lem4 21240 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  -u 1  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 )  e.  CC )
1412, 13mpan2 655 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1511, 14subcld 9125 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  CC )
16 ax-icn 8764 . . . . . . 7  |-  _i  e.  CC
171, 2, 3, 4, 5ipval2lem4 21240 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1816, 17mpan2 655 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1916negcli 9082 . . . . . . . . 9  |-  -u _i  e.  CC
201, 2, 3, 4, 5ipval2lem4 21240 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  -u _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
2119, 20mpan2 655 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
2218, 21subcld 9125 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )
23 mulcl 8789 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )  ->  ( _i  x.  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) )  e.  CC )
2416, 22, 23sylancr 647 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) )  e.  CC )
2515, 24addcld 8822 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  e.  CC )
26 4cn 9788 . . . . . 6  |-  4  e.  CC
27 4re 9787 . . . . . . 7  |-  4  e.  RR
28 4pos 9800 . . . . . . 7  |-  0  <  4
2927, 28gt0ne0ii 9277 . . . . . 6  |-  4  =/=  0
30 cjdiv 11615 . . . . . 6  |-  ( ( ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  e.  CC  /\  4  e.  CC  /\  4  =/=  0 )  ->  (
* `  ( (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )  =  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) ) )
3126, 29, 30mp3an23 1274 . . . . 5  |-  ( ( ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  e.  CC  ->  ( * `  ( ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )  =  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) ) )
3225, 31syl 17 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )  =  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) ) )
33 cjre 11590 . . . . . . 7  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
3427, 33ax-mp 10 . . . . . 6  |-  ( * `
 4 )  =  4
3534oveq2i 5803 . . . . 5  |-  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) )  =  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  4 )
361, 2, 3, 4, 5ipval2lem2 21238 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  -u 1  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 )  e.  RR )
3712, 36mpan2 655 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
3810, 37resubcld 9179 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  RR )
391, 2, 3, 4, 5ipval2lem2 21238 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
4016, 39mpan2 655 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
411, 2, 3, 4, 5ipval2lem2 21238 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  -u _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
4219, 41mpan2 655 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
4340, 42resubcld 9179 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  RR )
44 cjreim 11611 . . . . . . . 8  |-  ( ( ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  RR  /\  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  RR )  ->  ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  =  ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
4538, 43, 44syl2anc 645 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  =  ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
46 submul2 9188 . . . . . . . . 9  |-  ( ( ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  CC  /\  _i  e.  CC  /\  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )  ->  ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
4716, 46mp3an2 1270 . . . . . . . 8  |-  ( ( ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  CC  /\  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )  ->  ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
4815, 22, 47syl2anc 645 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
491, 2nvcom 21138 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) B )  =  ( B ( +v `  U ) A ) )
5049fveq2d 5462 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) B ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) A ) ) )
5150oveq1d 5807 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  =  ( ( ( normCV `  U ) `  ( B ( +v `  U ) A ) ) ^ 2 ) )
521, 2, 3, 4nvdif 21192 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) )  =  ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) )
5352oveq1d 5807 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )
5451, 53oveq12d 5810 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U ) `  ( B ( +v `  U ) A ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) ) )
5518, 21negsubdi2d 9141 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) )
561, 2, 3, 4nvpi 21193 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  (
( normCV `  U ) `  ( B ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) )  =  ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) )
57563com23 1162 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( B ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) )  =  ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) )
5857eqcomd 2263 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) )
5958oveq1d 5807 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 ) )
601, 2, 3, 4nvpi 21193 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) )
6160oveq1d 5807 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) )
6259, 61oveq12d 5810 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) )
6355, 62eqtrd 2290 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) )
6463oveq2d 5808 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i  x.  -u ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) )  =  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )
6554, 64oveq12d 5810 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( ( normCV `  U ) `  ( B ( +v `  U ) A ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) ) )
6645, 48, 653eqtrd 2294 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  =  ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) ) )
6766oveq1d 5807 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( * `  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  4 )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
6835, 67syl5eq 2302 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( * `  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
6932, 68eqtrd 2290 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
709, 69eqtr4d 2293 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( B P A )  =  ( * `  (
( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) ) )
717, 70eqtr4d 2293 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706   _ici 8707    + caddc 8708    x. cmul 8710    - cmin 9005   -ucneg 9006    / cdiv 9391   2c2 9763   4c4 9765   ^cexp 11071   *ccj 11547   NrmCVeccnv 21101   +vcpv 21102   BaseSetcba 21103   .s
OLDcns 21104   normCVcnmcv 21107   .i OLDcdip 21234
This theorem is referenced by:  ipipcj  21252  diporthcom  21253  dip0l  21255  ipasslem10  21378  dipdi  21382  dipassr  21385  dipsubdi  21388  siii  21392  hlipcj  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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-fz 10750  df-fzo 10838  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-sum 12125  df-grpo 20819  df-gid 20820  df-ginv 20821  df-ablo 20910  df-vc 21063  df-nv 21109  df-va 21112  df-ba 21113  df-sm 21114  df-0v 21115  df-nmcv 21117  df-dip 21235
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