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Theorem opsqrlem3 26737
Description: Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
opsqrlem2.1  |-  T  e. 
HrmOp
opsqrlem2.2  |-  S  =  ( x  e.  HrmOp ,  y  e.  HrmOp  |->  ( x 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )
opsqrlem2.3  |-  F  =  seq 1 ( S ,  ( NN  X.  { 0hop } ) )
Assertion
Ref Expression
opsqrlem3  |-  ( ( G  e.  HrmOp  /\  H  e.  HrmOp )  ->  ( G S H )  =  ( G  +op  (
( 1  /  2
)  .op  ( T  -op  ( G  o.  G
) ) ) ) )
Distinct variable group:    x, y, T
Allowed substitution hints:    S( x, y)    F( x, y)    G( x, y)    H( x, y)

Proof of Theorem opsqrlem3
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3  |-  ( z  =  G  ->  z  =  G )
21, 1coeq12d 5165 . . . . 5  |-  ( z  =  G  ->  (
z  o.  z )  =  ( G  o.  G ) )
32oveq2d 6298 . . . 4  |-  ( z  =  G  ->  ( T  -op  ( z  o.  z ) )  =  ( T  -op  ( G  o.  G )
) )
43oveq2d 6298 . . 3  |-  ( z  =  G  ->  (
( 1  /  2
)  .op  ( T  -op  ( z  o.  z
) ) )  =  ( ( 1  / 
2 )  .op  ( T  -op  ( G  o.  G ) ) ) )
51, 4oveq12d 6300 . 2  |-  ( z  =  G  ->  (
z  +op  ( (
1  /  2 ) 
.op  ( T  -op  ( z  o.  z
) ) ) )  =  ( G  +op  ( ( 1  / 
2 )  .op  ( T  -op  ( G  o.  G ) ) ) ) )
6 eqidd 2468 . 2  |-  ( w  =  H  ->  ( G  +op  ( ( 1  /  2 )  .op  ( T  -op  ( G  o.  G ) ) ) )  =  ( G  +op  ( ( 1  /  2 ) 
.op  ( T  -op  ( G  o.  G
) ) ) ) )
7 opsqrlem2.2 . . 3  |-  S  =  ( x  e.  HrmOp ,  y  e.  HrmOp  |->  ( x 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )
8 id 22 . . . . 5  |-  ( x  =  z  ->  x  =  z )
98, 8coeq12d 5165 . . . . . . 7  |-  ( x  =  z  ->  (
x  o.  x )  =  ( z  o.  z ) )
109oveq2d 6298 . . . . . 6  |-  ( x  =  z  ->  ( T  -op  ( x  o.  x ) )  =  ( T  -op  (
z  o.  z ) ) )
1110oveq2d 6298 . . . . 5  |-  ( x  =  z  ->  (
( 1  /  2
)  .op  ( T  -op  ( x  o.  x
) ) )  =  ( ( 1  / 
2 )  .op  ( T  -op  ( z  o.  z ) ) ) )
128, 11oveq12d 6300 . . . 4  |-  ( x  =  z  ->  (
x  +op  ( (
1  /  2 ) 
.op  ( T  -op  ( x  o.  x
) ) ) )  =  ( z  +op  ( ( 1  / 
2 )  .op  ( T  -op  ( z  o.  z ) ) ) ) )
13 eqidd 2468 . . . 4  |-  ( y  =  w  ->  (
z  +op  ( (
1  /  2 ) 
.op  ( T  -op  ( z  o.  z
) ) ) )  =  ( z  +op  ( ( 1  / 
2 )  .op  ( T  -op  ( z  o.  z ) ) ) ) )
1412, 13cbvmpt2v 6359 . . 3  |-  ( x  e.  HrmOp ,  y  e. 
HrmOp  |->  ( x  +op  ( ( 1  / 
2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )  =  ( z  e.  HrmOp ,  w  e.  HrmOp  |->  ( z  +op  ( ( 1  / 
2 )  .op  ( T  -op  ( z  o.  z ) ) ) ) )
157, 14eqtri 2496 . 2  |-  S  =  ( z  e.  HrmOp ,  w  e.  HrmOp  |->  ( z 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( z  o.  z ) ) ) ) )
16 ovex 6307 . 2  |-  ( G 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( G  o.  G ) ) ) )  e.  _V
175, 6, 15, 16ovmpt2 6420 1  |-  ( ( G  e.  HrmOp  /\  H  e.  HrmOp )  ->  ( G S H )  =  ( G  +op  (
( 1  /  2
)  .op  ( T  -op  ( G  o.  G
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4027    X. cxp 4997    o. ccom 5003  (class class class)co 6282    |-> cmpt2 6284   1c1 9489    / cdiv 10202   NNcn 10532   2c2 10581    seqcseq 12071    +op chos 25531    .op chot 25532    -op chod 25533   0hopch0o 25536   HrmOpcho 25543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287
This theorem is referenced by:  opsqrlem4  26738  opsqrlem5  26739
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