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Theorem opsqrlem3 23645
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 20 . . 3  |-  ( z  =  G  ->  z  =  G )
21, 1coeq12d 5037 . . . . 5  |-  ( z  =  G  ->  (
z  o.  z )  =  ( G  o.  G ) )
32oveq2d 6097 . . . 4  |-  ( z  =  G  ->  ( T  -op  ( z  o.  z ) )  =  ( T  -op  ( G  o.  G )
) )
43oveq2d 6097 . . 3  |-  ( z  =  G  ->  (
( 1  /  2
)  .op  ( T  -op  ( z  o.  z
) ) )  =  ( ( 1  / 
2 )  .op  ( T  -op  ( G  o.  G ) ) ) )
51, 4oveq12d 6099 . 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 2437 . 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 20 . . . . 5  |-  ( x  =  z  ->  x  =  z )
98, 8coeq12d 5037 . . . . . . 7  |-  ( x  =  z  ->  (
x  o.  x )  =  ( z  o.  z ) )
109oveq2d 6097 . . . . . 6  |-  ( x  =  z  ->  ( T  -op  ( x  o.  x ) )  =  ( T  -op  (
z  o.  z ) ) )
1110oveq2d 6097 . . . . 5  |-  ( x  =  z  ->  (
( 1  /  2
)  .op  ( T  -op  ( x  o.  x
) ) )  =  ( ( 1  / 
2 )  .op  ( T  -op  ( z  o.  z ) ) ) )
128, 11oveq12d 6099 . . . 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 2437 . . . 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 6152 . . 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 2456 . 2  |-  S  =  ( z  e.  HrmOp ,  w  e.  HrmOp  |->  ( z 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( z  o.  z ) ) ) ) )
16 ovex 6106 . 2  |-  ( G 
+op  ( ( 1  /  2 )  .op  ( T  -op  ( G  o.  G ) ) ) )  e.  _V
175, 6, 15, 16ovmpt2 6209 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 359    = wceq 1652    e. wcel 1725   {csn 3814    X. cxp 4876    o. ccom 4882  (class class class)co 6081    e. cmpt2 6083   1c1 8991    / cdiv 9677   NNcn 10000   2c2 10049    seq cseq 11323    +op chos 22441    .op chot 22442    -op chod 22443   0hopch0o 22446   HrmOpcho 22453
This theorem is referenced by:  opsqrlem4  23646  opsqrlem5  23647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086
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