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Theorem opsqrlem3 22722
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 19 . . 3  |-  ( z  =  G  ->  z  =  G )
21, 1coeq12d 4848 . . . . 5  |-  ( z  =  G  ->  (
z  o.  z )  =  ( G  o.  G ) )
32oveq2d 5874 . . . 4  |-  ( z  =  G  ->  ( T  -op  ( z  o.  z ) )  =  ( T  -op  ( G  o.  G )
) )
43oveq2d 5874 . . 3  |-  ( z  =  G  ->  (
( 1  /  2
)  .op  ( T  -op  ( z  o.  z
) ) )  =  ( ( 1  / 
2 )  .op  ( T  -op  ( G  o.  G ) ) ) )
51, 4oveq12d 5876 . 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 2284 . 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 19 . . . . 5  |-  ( x  =  z  ->  x  =  z )
98, 8coeq12d 4848 . . . . . . 7  |-  ( x  =  z  ->  (
x  o.  x )  =  ( z  o.  z ) )
109oveq2d 5874 . . . . . 6  |-  ( x  =  z  ->  ( T  -op  ( x  o.  x ) )  =  ( T  -op  (
z  o.  z ) ) )
1110oveq2d 5874 . . . . 5  |-  ( x  =  z  ->  (
( 1  /  2
)  .op  ( T  -op  ( x  o.  x
) ) )  =  ( ( 1  / 
2 )  .op  ( T  -op  ( z  o.  z ) ) ) )
128, 11oveq12d 5876 . . . 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 2284 . . . 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 5926 . . 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 2303 . 2  |-  S  =  ( z  e.  HrmOp ,  w  e.  HrmOp  |->  ( z 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( z  o.  z ) ) ) ) )
16 ovex 5883 . 2  |-  ( G 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( G  o.  G ) ) ) )  e.  _V
175, 6, 15, 16ovmpt2 5983 1  |-  ( ( G  e.  HrmOp  /\  H  e.  HrmOp )  ->  ( G S H )  =  ( G  +op  (
( 1  /  2
)  .op  ( T  -op  ( G  o.  G
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    X. cxp 4687    o. ccom 4693  (class class class)co 5858    e. cmpt2 5860   1c1 8738    / cdiv 9423   NNcn 9746   2c2 9795    seq cseq 11046    +op chos 21518    .op chot 21519    -op chod 21520   0hopch0o 21523   HrmOpcho 21530
This theorem is referenced by:  opsqrlem4  22723  opsqrlem5  22724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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