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Theorem ipdirilem 22291
Description: Lemma for ipdiri 22292. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1  |-  X  =  ( BaseSet `  U )
ip1i.2  |-  G  =  ( +v `  U
)
ip1i.4  |-  S  =  ( .s OLD `  U
)
ip1i.7  |-  P  =  ( .i OLD `  U
)
ip1i.9  |-  U  e.  CPreHil
OLD
ipdiri.8  |-  A  e.  X
ipdiri.9  |-  B  e.  X
ipdiri.10  |-  C  e.  X
Assertion
Ref Expression
ipdirilem  |-  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) )

Proof of Theorem ipdirilem
StepHypRef Expression
1 2cn 10034 . . . . . . 7  |-  2  e.  CC
2 2ne0 10047 . . . . . . 7  |-  2  =/=  0
31, 2recidi 9709 . . . . . 6  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
43oveq1i 6058 . . . . 5  |-  ( ( 2  x.  ( 1  /  2 ) ) S ( A G B ) )  =  ( 1 S ( A G B ) )
5 ip1i.9 . . . . . . 7  |-  U  e.  CPreHil
OLD
65phnvi 22278 . . . . . 6  |-  U  e.  NrmCVec
71, 2reccli 9708 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
8 ipdiri.8 . . . . . . . 8  |-  A  e.  X
9 ipdiri.9 . . . . . . . 8  |-  B  e.  X
10 ip1i.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
11 ip1i.2 . . . . . . . . 9  |-  G  =  ( +v `  U
)
1210, 11nvgcl 22060 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
136, 8, 9, 12mp3an 1279 . . . . . . 7  |-  ( A G B )  e.  X
141, 7, 133pm3.2i 1132 . . . . . 6  |-  ( 2  e.  CC  /\  (
1  /  2 )  e.  CC  /\  ( A G B )  e.  X )
15 ip1i.4 . . . . . . 7  |-  S  =  ( .s OLD `  U
)
1610, 15nvsass 22070 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  (
2  e.  CC  /\  ( 1  /  2
)  e.  CC  /\  ( A G B )  e.  X ) )  ->  ( ( 2  x.  ( 1  / 
2 ) ) S ( A G B ) )  =  ( 2 S ( ( 1  /  2 ) S ( A G B ) ) ) )
176, 14, 16mp2an 654 . . . . 5  |-  ( ( 2  x.  ( 1  /  2 ) ) S ( A G B ) )  =  ( 2 S ( ( 1  /  2
) S ( A G B ) ) )
1810, 15nvsid 22069 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A G B )  e.  X )  ->  (
1 S ( A G B ) )  =  ( A G B ) )
196, 13, 18mp2an 654 . . . . 5  |-  ( 1 S ( A G B ) )  =  ( A G B )
204, 17, 193eqtr3i 2440 . . . 4  |-  ( 2 S ( ( 1  /  2 ) S ( A G B ) ) )  =  ( A G B )
2120oveq1i 6058 . . 3  |-  ( ( 2 S ( ( 1  /  2 ) S ( A G B ) ) ) P C )  =  ( ( A G B ) P C )
22 ip1i.7 . . . 4  |-  P  =  ( .i OLD `  U
)
2310, 15nvscl 22068 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  (
1  /  2 )  e.  CC  /\  ( A G B )  e.  X )  ->  (
( 1  /  2
) S ( A G B ) )  e.  X )
246, 7, 13, 23mp3an 1279 . . . 4  |-  ( ( 1  /  2 ) S ( A G B ) )  e.  X
25 ipdiri.10 . . . 4  |-  C  e.  X
2610, 11, 15, 22, 5, 24, 25ip2i 22290 . . 3  |-  ( ( 2 S ( ( 1  /  2 ) S ( A G B ) ) ) P C )  =  ( 2  x.  (
( ( 1  / 
2 ) S ( A G B ) ) P C ) )
2721, 26eqtr3i 2434 . 2  |-  ( ( A G B ) P C )  =  ( 2  x.  (
( ( 1  / 
2 ) S ( A G B ) ) P C ) )
28 neg1cn 10031 . . . . . 6  |-  -u 1  e.  CC
2910, 15nvscl 22068 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 S B )  e.  X )
306, 28, 9, 29mp3an 1279 . . . . 5  |-  ( -u
1 S B )  e.  X
3110, 11nvgcl 22060 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( -u 1 S B )  e.  X )  -> 
( A G (
-u 1 S B ) )  e.  X
)
326, 8, 30, 31mp3an 1279 . . . 4  |-  ( A G ( -u 1 S B ) )  e.  X
3310, 15nvscl 22068 . . . 4  |-  ( ( U  e.  NrmCVec  /\  (
1  /  2 )  e.  CC  /\  ( A G ( -u 1 S B ) )  e.  X )  ->  (
( 1  /  2
) S ( A G ( -u 1 S B ) ) )  e.  X )
346, 7, 32, 33mp3an 1279 . . 3  |-  ( ( 1  /  2 ) S ( A G ( -u 1 S B ) ) )  e.  X
3510, 11, 15, 22, 5, 24, 34, 25ip1i 22289 . 2  |-  ( ( ( ( ( 1  /  2 ) S ( A G B ) ) G ( ( 1  /  2
) S ( A G ( -u 1 S B ) ) ) ) P C )  +  ( ( ( ( 1  /  2
) S ( A G B ) ) G ( -u 1 S ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) ) ) P C ) )  =  ( 2  x.  ( ( ( 1  /  2 ) S ( A G B ) ) P C ) )
36 eqid 2412 . . . . . . . . . . . 12  |-  ( 1st `  U )  =  ( 1st `  U )
3736nvvc 22055 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
386, 37ax-mp 8 . . . . . . . . . 10  |-  ( 1st `  U )  e.  CVec OLD
3911vafval 22043 . . . . . . . . . . 11  |-  G  =  ( 1st `  ( 1st `  U ) )
4039vcablo 21997 . . . . . . . . . 10  |-  ( ( 1st `  U )  e.  CVec OLD  ->  G  e. 
AbelOp )
4138, 40ax-mp 8 . . . . . . . . 9  |-  G  e. 
AbelOp
428, 9pm3.2i 442 . . . . . . . . 9  |-  ( A  e.  X  /\  B  e.  X )
438, 30pm3.2i 442 . . . . . . . . 9  |-  ( A  e.  X  /\  ( -u 1 S B )  e.  X )
4410, 11bafval 22044 . . . . . . . . . 10  |-  X  =  ran  G
4544ablo4 21836 . . . . . . . . 9  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X )  /\  ( A  e.  X  /\  ( -u 1 S B )  e.  X
) )  ->  (
( A G B ) G ( A G ( -u 1 S B ) ) )  =  ( ( A G A ) G ( B G (
-u 1 S B ) ) ) )
4641, 42, 43, 45mp3an 1279 . . . . . . . 8  |-  ( ( A G B ) G ( A G ( -u 1 S B ) ) )  =  ( ( A G A ) G ( B G (
-u 1 S B ) ) )
4715smfval 22045 . . . . . . . . . . 11  |-  S  =  ( 2nd `  ( 1st `  U ) )
4839, 47, 44vc2 21995 . . . . . . . . . 10  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
4938, 8, 48mp2an 654 . . . . . . . . 9  |-  ( A G A )  =  ( 2 S A )
50 eqid 2412 . . . . . . . . . . 11  |-  ( 0vec `  U )  =  (
0vec `  U )
5110, 11, 15, 50nvrinv 22095 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B G ( -u 1 S B ) )  =  ( 0vec `  U
) )
526, 9, 51mp2an 654 . . . . . . . . 9  |-  ( B G ( -u 1 S B ) )  =  ( 0vec `  U
)
5349, 52oveq12i 6060 . . . . . . . 8  |-  ( ( A G A ) G ( B G ( -u 1 S B ) ) )  =  ( ( 2 S A ) G ( 0vec `  U
) )
5410, 15nvscl 22068 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  2  e.  CC  /\  A  e.  X )  ->  (
2 S A )  e.  X )
556, 1, 8, 54mp3an 1279 . . . . . . . . 9  |-  ( 2 S A )  e.  X
5610, 11, 50nv0rid 22077 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  (
2 S A )  e.  X )  -> 
( ( 2 S A ) G (
0vec `  U )
)  =  ( 2 S A ) )
576, 55, 56mp2an 654 . . . . . . . 8  |-  ( ( 2 S A ) G ( 0vec `  U
) )  =  ( 2 S A )
5846, 53, 573eqtri 2436 . . . . . . 7  |-  ( ( A G B ) G ( A G ( -u 1 S B ) ) )  =  ( 2 S A )
5958oveq2i 6059 . . . . . 6  |-  ( ( 1  /  2 ) S ( ( A G B ) G ( A G (
-u 1 S B ) ) ) )  =  ( ( 1  /  2 ) S ( 2 S A ) )
607, 1, 83pm3.2i 1132 . . . . . . 7  |-  ( ( 1  /  2 )  e.  CC  /\  2  e.  CC  /\  A  e.  X )
6110, 15nvsass 22070 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  2
)  e.  CC  /\  2  e.  CC  /\  A  e.  X ) )  -> 
( ( ( 1  /  2 )  x.  2 ) S A )  =  ( ( 1  /  2 ) S ( 2 S A ) ) )
626, 60, 61mp2an 654 . . . . . 6  |-  ( ( ( 1  /  2
)  x.  2 ) S A )  =  ( ( 1  / 
2 ) S ( 2 S A ) )
6359, 62eqtr4i 2435 . . . . 5  |-  ( ( 1  /  2 ) S ( ( A G B ) G ( A G (
-u 1 S B ) ) ) )  =  ( ( ( 1  /  2 )  x.  2 ) S A )
647, 13, 323pm3.2i 1132 . . . . . 6  |-  ( ( 1  /  2 )  e.  CC  /\  ( A G B )  e.  X  /\  ( A G ( -u 1 S B ) )  e.  X )
6510, 11, 15nvdi 22072 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  2
)  e.  CC  /\  ( A G B )  e.  X  /\  ( A G ( -u 1 S B ) )  e.  X ) )  -> 
( ( 1  / 
2 ) S ( ( A G B ) G ( A G ( -u 1 S B ) ) ) )  =  ( ( ( 1  /  2
) S ( A G B ) ) G ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) ) )
666, 64, 65mp2an 654 . . . . 5  |-  ( ( 1  /  2 ) S ( ( A G B ) G ( A G (
-u 1 S B ) ) ) )  =  ( ( ( 1  /  2 ) S ( A G B ) ) G ( ( 1  / 
2 ) S ( A G ( -u
1 S B ) ) ) )
67 ax-1cn 9012 . . . . . . . 8  |-  1  e.  CC
6867, 1, 2divcan1i 9722 . . . . . . 7  |-  ( ( 1  /  2 )  x.  2 )  =  1
6968oveq1i 6058 . . . . . 6  |-  ( ( ( 1  /  2
)  x.  2 ) S A )  =  ( 1 S A )
7010, 15nvsid 22069 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
716, 8, 70mp2an 654 . . . . . 6  |-  ( 1 S A )  =  A
7269, 71eqtri 2432 . . . . 5  |-  ( ( ( 1  /  2
)  x.  2 ) S A )  =  A
7363, 66, 723eqtr3i 2440 . . . 4  |-  ( ( ( 1  /  2
) S ( A G B ) ) G ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) )  =  A
7473oveq1i 6058 . . 3  |-  ( ( ( ( 1  / 
2 ) S ( A G B ) ) G ( ( 1  /  2 ) S ( A G ( -u 1 S B ) ) ) ) P C )  =  ( A P C )
7528, 7mulcomi 9060 . . . . . . . . 9  |-  ( -u
1  x.  ( 1  /  2 ) )  =  ( ( 1  /  2 )  x.  -u 1 )
7675oveq1i 6058 . . . . . . . 8  |-  ( (
-u 1  x.  (
1  /  2 ) ) S ( A G ( -u 1 S B ) ) )  =  ( ( ( 1  /  2 )  x.  -u 1 ) S ( A G (
-u 1 S B ) ) )
7728, 7, 323pm3.2i 1132 . . . . . . . . 9  |-  ( -u
1  e.  CC  /\  ( 1  /  2
)  e.  CC  /\  ( A G ( -u
1 S B ) )  e.  X )
7810, 15nvsass 22070 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  ( 1  /  2
)  e.  CC  /\  ( A G ( -u
1 S B ) )  e.  X ) )  ->  ( ( -u 1  x.  ( 1  /  2 ) ) S ( A G ( -u 1 S B ) ) )  =  ( -u 1 S ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) ) )
796, 77, 78mp2an 654 . . . . . . . 8  |-  ( (
-u 1  x.  (
1  /  2 ) ) S ( A G ( -u 1 S B ) ) )  =  ( -u 1 S ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) )
807, 28, 323pm3.2i 1132 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  CC  /\  -u 1  e.  CC  /\  ( A G ( -u 1 S B ) )  e.  X )
8110, 15nvsass 22070 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  2
)  e.  CC  /\  -u 1  e.  CC  /\  ( A G ( -u
1 S B ) )  e.  X ) )  ->  ( (
( 1  /  2
)  x.  -u 1
) S ( A G ( -u 1 S B ) ) )  =  ( ( 1  /  2 ) S ( -u 1 S ( A G (
-u 1 S B ) ) ) ) )
826, 80, 81mp2an 654 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  x.  -u 1
) S ( A G ( -u 1 S B ) ) )  =  ( ( 1  /  2 ) S ( -u 1 S ( A G (
-u 1 S B ) ) ) )
8328, 8, 303pm3.2i 1132 . . . . . . . . . . . 12  |-  ( -u
1  e.  CC  /\  A  e.  X  /\  ( -u 1 S B )  e.  X )
8410, 11, 15nvdi 22072 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  A  e.  X  /\  ( -u 1 S B )  e.  X ) )  ->  ( -u 1 S ( A G ( -u 1 S B ) ) )  =  ( ( -u
1 S A ) G ( -u 1 S ( -u 1 S B ) ) ) )
856, 83, 84mp2an 654 . . . . . . . . . . 11  |-  ( -u
1 S ( A G ( -u 1 S B ) ) )  =  ( ( -u
1 S A ) G ( -u 1 S ( -u 1 S B ) ) )
8667, 67mul2negi 9445 . . . . . . . . . . . . . . 15  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
87 1t1e1 10090 . . . . . . . . . . . . . . 15  |-  ( 1  x.  1 )  =  1
8886, 87eqtri 2432 . . . . . . . . . . . . . 14  |-  ( -u
1  x.  -u 1
)  =  1
8988oveq1i 6058 . . . . . . . . . . . . 13  |-  ( (
-u 1  x.  -u 1
) S B )  =  ( 1 S B )
9028, 28, 93pm3.2i 1132 . . . . . . . . . . . . . 14  |-  ( -u
1  e.  CC  /\  -u 1  e.  CC  /\  B  e.  X )
9110, 15nvsass 22070 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  -u 1  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  -u 1
) S B )  =  ( -u 1 S ( -u 1 S B ) ) )
926, 90, 91mp2an 654 . . . . . . . . . . . . 13  |-  ( (
-u 1  x.  -u 1
) S B )  =  ( -u 1 S ( -u 1 S B ) )
9310, 15nvsid 22069 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
946, 9, 93mp2an 654 . . . . . . . . . . . . 13  |-  ( 1 S B )  =  B
9589, 92, 943eqtr3i 2440 . . . . . . . . . . . 12  |-  ( -u
1 S ( -u
1 S B ) )  =  B
9695oveq2i 6059 . . . . . . . . . . 11  |-  ( (
-u 1 S A ) G ( -u
1 S ( -u
1 S B ) ) )  =  ( ( -u 1 S A ) G B )
9785, 96eqtri 2432 . . . . . . . . . 10  |-  ( -u
1 S ( A G ( -u 1 S B ) ) )  =  ( ( -u
1 S A ) G B )
9897oveq2i 6059 . . . . . . . . 9  |-  ( ( 1  /  2 ) S ( -u 1 S ( A G ( -u 1 S B ) ) ) )  =  ( ( 1  /  2 ) S ( ( -u
1 S A ) G B ) )
9982, 98eqtri 2432 . . . . . . . 8  |-  ( ( ( 1  /  2
)  x.  -u 1
) S ( A G ( -u 1 S B ) ) )  =  ( ( 1  /  2 ) S ( ( -u 1 S A ) G B ) )
10076, 79, 993eqtr3i 2440 . . . . . . 7  |-  ( -u
1 S ( ( 1  /  2 ) S ( A G ( -u 1 S B ) ) ) )  =  ( ( 1  /  2 ) S ( ( -u
1 S A ) G B ) )
101100oveq2i 6059 . . . . . 6  |-  ( ( ( 1  /  2
) S ( A G B ) ) G ( -u 1 S ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) ) )  =  ( ( ( 1  /  2
) S ( A G B ) ) G ( ( 1  /  2 ) S ( ( -u 1 S A ) G B ) ) )
10210, 15nvscl 22068 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
1036, 28, 8, 102mp3an 1279 . . . . . . . . 9  |-  ( -u
1 S A )  e.  X
10410, 11nvgcl 22060 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( -u 1 S A )  e.  X  /\  B  e.  X )  ->  (
( -u 1 S A ) G B )  e.  X )
1056, 103, 9, 104mp3an 1279 . . . . . . . 8  |-  ( (
-u 1 S A ) G B )  e.  X
1067, 13, 1053pm3.2i 1132 . . . . . . 7  |-  ( ( 1  /  2 )  e.  CC  /\  ( A G B )  e.  X  /\  ( (
-u 1 S A ) G B )  e.  X )
10710, 11, 15nvdi 22072 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  2
)  e.  CC  /\  ( A G B )  e.  X  /\  (
( -u 1 S A ) G B )  e.  X ) )  ->  ( ( 1  /  2 ) S ( ( A G B ) G ( ( -u 1 S A ) G B ) ) )  =  ( ( ( 1  /  2 ) S ( A G B ) ) G ( ( 1  /  2
) S ( (
-u 1 S A ) G B ) ) ) )
1086, 106, 107mp2an 654 . . . . . 6  |-  ( ( 1  /  2 ) S ( ( A G B ) G ( ( -u 1 S A ) G B ) ) )  =  ( ( ( 1  /  2 ) S ( A G B ) ) G ( ( 1  /  2
) S ( (
-u 1 S A ) G B ) ) )
109101, 108eqtr4i 2435 . . . . 5  |-  ( ( ( 1  /  2
) S ( A G B ) ) G ( -u 1 S ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) ) )  =  ( ( 1  /  2 ) S ( ( A G B ) G ( ( -u 1 S A ) G B ) ) )
110103, 9pm3.2i 442 . . . . . . . . 9  |-  ( (
-u 1 S A )  e.  X  /\  B  e.  X )
11144ablo4 21836 . . . . . . . . 9  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X )  /\  ( ( -u 1 S A )  e.  X  /\  B  e.  X
) )  ->  (
( A G B ) G ( (
-u 1 S A ) G B ) )  =  ( ( A G ( -u
1 S A ) ) G ( B G B ) ) )
11241, 42, 110, 111mp3an 1279 . . . . . . . 8  |-  ( ( A G B ) G ( ( -u
1 S A ) G B ) )  =  ( ( A G ( -u 1 S A ) ) G ( B G B ) )
11310, 11, 15, 50nvrinv 22095 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
1146, 8, 113mp2an 654 . . . . . . . . . 10  |-  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
)
115114oveq1i 6058 . . . . . . . . 9  |-  ( ( A G ( -u
1 S A ) ) G ( B G B ) )  =  ( ( 0vec `  U ) G ( B G B ) )
11610, 11nvgcl 22060 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  B  e.  X )  ->  ( B G B )  e.  X )
1176, 9, 9, 116mp3an 1279 . . . . . . . . . 10  |-  ( B G B )  e.  X
11810, 11, 50nv0lid 22078 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  ( B G B )  e.  X )  ->  (
( 0vec `  U ) G ( B G B ) )  =  ( B G B ) )
1196, 117, 118mp2an 654 . . . . . . . . 9  |-  ( (
0vec `  U ) G ( B G B ) )  =  ( B G B )
120115, 119eqtri 2432 . . . . . . . 8  |-  ( ( A G ( -u
1 S A ) ) G ( B G B ) )  =  ( B G B )
12139, 47, 44vc2 21995 . . . . . . . . 9  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  B  e.  X )  ->  ( B G B )  =  ( 2 S B ) )
12238, 9, 121mp2an 654 . . . . . . . 8  |-  ( B G B )  =  ( 2 S B )
123112, 120, 1223eqtri 2436 . . . . . . 7  |-  ( ( A G B ) G ( ( -u
1 S A ) G B ) )  =  ( 2 S B )
124123oveq2i 6059 . . . . . 6  |-  ( ( 1  /  2 ) S ( ( A G B ) G ( ( -u 1 S A ) G B ) ) )  =  ( ( 1  / 
2 ) S ( 2 S B ) )
1257, 1, 93pm3.2i 1132 . . . . . . 7  |-  ( ( 1  /  2 )  e.  CC  /\  2  e.  CC  /\  B  e.  X )
12610, 15nvsass 22070 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  2
)  e.  CC  /\  2  e.  CC  /\  B  e.  X ) )  -> 
( ( ( 1  /  2 )  x.  2 ) S B )  =  ( ( 1  /  2 ) S ( 2 S B ) ) )
1276, 125, 126mp2an 654 . . . . . 6  |-  ( ( ( 1  /  2
)  x.  2 ) S B )  =  ( ( 1  / 
2 ) S ( 2 S B ) )
12868oveq1i 6058 . . . . . 6  |-  ( ( ( 1  /  2
)  x.  2 ) S B )  =  ( 1 S B )
129124, 127, 1283eqtr2i 2438 . . . . 5  |-  ( ( 1  /  2 ) S ( ( A G B ) G ( ( -u 1 S A ) G B ) ) )  =  ( 1 S B )
130109, 129, 943eqtri 2436 . . . 4  |-  ( ( ( 1  /  2
) S ( A G B ) ) G ( -u 1 S ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) ) )  =  B
131130oveq1i 6058 . . 3  |-  ( ( ( ( 1  / 
2 ) S ( A G B ) ) G ( -u
1 S ( ( 1  /  2 ) S ( A G ( -u 1 S B ) ) ) ) ) P C )  =  ( B P C )
13274, 131oveq12i 6060 . 2  |-  ( ( ( ( ( 1  /  2 ) S ( A G B ) ) G ( ( 1  /  2
) S ( A G ( -u 1 S B ) ) ) ) P C )  +  ( ( ( ( 1  /  2
) S ( A G B ) ) G ( -u 1 S ( ( 1  /  2 ) S ( A G (
-u 1 S B ) ) ) ) ) P C ) )  =  ( ( A P C )  +  ( B P C ) )
13327, 35, 1323eqtr2i 2438 1  |-  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   1stc1st 6314   CCcc 8952   1c1 8955    + caddc 8957    x. cmul 8959   -ucneg 9256    / cdiv 9641   2c2 10013   AbelOpcablo 21830   CVec
OLDcvc 21985   NrmCVeccnv 22024   +vcpv 22025   BaseSetcba 22026   .s
OLDcns 22027   0veccn0v 22028   .i
OLDcdip 22157   CPreHil OLDccphlo 22274
This theorem is referenced by:  ipdiri  22292
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-grpo 21740  df-gid 21741  df-ginv 21742  df-ablo 21831  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-nmcv 22040  df-dip 22158  df-ph 22275
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