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Theorem opsqrlem3 27460
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 4987 . . . . 5  |-  ( z  =  G  ->  (
z  o.  z )  =  ( G  o.  G ) )
32oveq2d 6293 . . . 4  |-  ( z  =  G  ->  ( T  -op  ( z  o.  z ) )  =  ( T  -op  ( G  o.  G )
) )
43oveq2d 6293 . . 3  |-  ( z  =  G  ->  (
( 1  /  2
)  .op  ( T  -op  ( z  o.  z
) ) )  =  ( ( 1  / 
2 )  .op  ( T  -op  ( G  o.  G ) ) ) )
51, 4oveq12d 6295 . 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 2403 . 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 4987 . . . . . . 7  |-  ( x  =  z  ->  (
x  o.  x )  =  ( z  o.  z ) )
109oveq2d 6293 . . . . . 6  |-  ( x  =  z  ->  ( T  -op  ( x  o.  x ) )  =  ( T  -op  (
z  o.  z ) ) )
1110oveq2d 6293 . . . . 5  |-  ( x  =  z  ->  (
( 1  /  2
)  .op  ( T  -op  ( x  o.  x
) ) )  =  ( ( 1  / 
2 )  .op  ( T  -op  ( z  o.  z ) ) ) )
128, 11oveq12d 6295 . . . 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 2403 . . . 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 6357 . . 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 2431 . 2  |-  S  =  ( z  e.  HrmOp ,  w  e.  HrmOp  |->  ( z 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( z  o.  z ) ) ) ) )
16 ovex 6305 . 2  |-  ( G 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( G  o.  G ) ) ) )  e.  _V
175, 6, 15, 16ovmpt2 6418 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 367    = wceq 1405    e. wcel 1842   {csn 3971    X. cxp 4820    o. ccom 4826  (class class class)co 6277    |-> cmpt2 6279   1c1 9522    / cdiv 10246   NNcn 10575   2c2 10625    seqcseq 12149    +op chos 26255    .op chot 26256    -op chod 26257   0hopch0o 26260   HrmOpcho 26267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282
This theorem is referenced by:  opsqrlem4  27461  opsqrlem5  27462
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