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Theorem psgnghm 27414
Description: The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
Hypotheses
Ref Expression
psgnghm.s  |-  S  =  ( SymGrp `  D )
psgnghm.n  |-  N  =  (pmSgn `  D )
psgnghm.f  |-  F  =  ( Ss  dom  N )
psgnghm.u  |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
Assertion
Ref Expression
psgnghm  |-  ( D  e.  V  ->  N  e.  ( F  GrpHom  U ) )

Proof of Theorem psgnghm
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnghm.s . . . . . 6  |-  S  =  ( SymGrp `  D )
2 eqid 2436 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2436 . . . . . 6  |-  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  =  {
x  e.  ( Base `  S )  |  dom  ( x  \  _I  )  e.  Fin }
4 psgnghm.n . . . . . 6  |-  N  =  (pmSgn `  D )
51, 2, 3, 4psgnfn 27401 . . . . 5  |-  N  Fn  { x  e.  ( Base `  S )  |  dom  ( x  \  _I  )  e.  Fin }
6 fndm 5544 . . . . 5  |-  ( N  Fn  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  ->  dom  N  =  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin } )
75, 6ax-mp 8 . . . 4  |-  dom  N  =  { x  e.  (
Base `  S )  |  dom  ( x  \  _I  )  e.  Fin }
8 ssrab2 3428 . . . 4  |-  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  C_  ( Base `  S )
97, 8eqsstri 3378 . . 3  |-  dom  N  C_  ( Base `  S
)
10 psgnghm.f . . . 4  |-  F  =  ( Ss  dom  N )
1110, 2ressbas2 13520 . . 3  |-  ( dom 
N  C_  ( Base `  S )  ->  dom  N  =  ( Base `  F
) )
129, 11ax-mp 8 . 2  |-  dom  N  =  ( Base `  F
)
13 psgnghm.u . . 3  |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
1413cnmsgnbas 27412 . 2  |-  { 1 ,  -u 1 }  =  ( Base `  U )
15 fvex 5742 . . . 4  |-  ( Base `  F )  e.  _V
1612, 15eqeltri 2506 . . 3  |-  dom  N  e.  _V
17 eqid 2436 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
1810, 17ressplusg 13571 . . 3  |-  ( dom 
N  e.  _V  ->  ( +g  `  S )  =  ( +g  `  F
) )
1916, 18ax-mp 8 . 2  |-  ( +g  `  S )  =  ( +g  `  F )
20 prex 4406 . . 3  |-  { 1 ,  -u 1 }  e.  _V
21 eqid 2436 . . . . 5  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
22 cnfldmul 16709 . . . . 5  |-  x.  =  ( .r ` fld )
2321, 22mgpplusg 15652 . . . 4  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2413, 23ressplusg 13571 . . 3  |-  ( { 1 ,  -u 1 }  e.  _V  ->  x.  =  ( +g  `  U
) )
2520, 24ax-mp 8 . 2  |-  x.  =  ( +g  `  U )
261, 4psgndmsubg 27402 . . 3  |-  ( D  e.  V  ->  dom  N  e.  (SubGrp `  S
) )
2710subggrp 14947 . . 3  |-  ( dom 
N  e.  (SubGrp `  S )  ->  F  e.  Grp )
2826, 27syl 16 . 2  |-  ( D  e.  V  ->  F  e.  Grp )
2913cnmsgngrp 27413 . . 3  |-  U  e. 
Grp
3029a1i 11 . 2  |-  ( D  e.  V  ->  U  e.  Grp )
31 fnfun 5542 . . . . . 6  |-  ( N  Fn  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  ->  Fun  N )
325, 31ax-mp 8 . . . . 5  |-  Fun  N
33 funfn 5482 . . . . 5  |-  ( Fun 
N  <->  N  Fn  dom  N )
3432, 33mpbi 200 . . . 4  |-  N  Fn  dom  N
3534a1i 11 . . 3  |-  ( D  e.  V  ->  N  Fn  dom  N )
36 eqid 2436 . . . . . 6  |-  ran  (pmTrsp `  D )  =  ran  (pmTrsp `  D )
371, 36, 4psgnvali 27408 . . . . 5  |-  ( x  e.  dom  N  ->  E. z  e. Word  ran  (pmTrsp `  D ) ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) ) )
38 lencl 11735 . . . . . . . . . . 11  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  z
)  e.  NN0 )
3938nn0zd 10373 . . . . . . . . . 10  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  z
)  e.  ZZ )
40 m1expcl2 11403 . . . . . . . . . . 11  |-  ( (
# `  z )  e.  ZZ  ->  ( -u 1 ^ ( # `  z
) )  e.  { -u 1 ,  1 } )
41 prcom 3882 . . . . . . . . . . 11  |-  { -u
1 ,  1 }  =  { 1 , 
-u 1 }
4240, 41syl6eleq 2526 . . . . . . . . . 10  |-  ( (
# `  z )  e.  ZZ  ->  ( -u 1 ^ ( # `  z
) )  e.  {
1 ,  -u 1 } )
4339, 42syl 16 . . . . . . . . 9  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( -u 1 ^ ( # `  z
) )  e.  {
1 ,  -u 1 } )
4443adantl 453 . . . . . . . 8  |-  ( ( D  e.  V  /\  z  e. Word  ran  (pmTrsp `  D ) )  -> 
( -u 1 ^ ( # `
 z ) )  e.  { 1 , 
-u 1 } )
45 eleq1a 2505 . . . . . . . 8  |-  ( (
-u 1 ^ ( # `
 z ) )  e.  { 1 , 
-u 1 }  ->  ( ( N `  x
)  =  ( -u
1 ^ ( # `  z ) )  -> 
( N `  x
)  e.  { 1 ,  -u 1 } ) )
4644, 45syl 16 . . . . . . 7  |-  ( ( D  e.  V  /\  z  e. Word  ran  (pmTrsp `  D ) )  -> 
( ( N `  x )  =  (
-u 1 ^ ( # `
 z ) )  ->  ( N `  x )  e.  {
1 ,  -u 1 } ) )
4746adantld 454 . . . . . 6  |-  ( ( D  e.  V  /\  z  e. Word  ran  (pmTrsp `  D ) )  -> 
( ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  ( -u 1 ^ ( # `  z
) ) )  -> 
( N `  x
)  e.  { 1 ,  -u 1 } ) )
4847rexlimdva 2830 . . . . 5  |-  ( D  e.  V  ->  ( E. z  e. Word  ran  (pmTrsp `  D ) ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) )  ->  ( N `  x )  e.  {
1 ,  -u 1 } ) )
4937, 48syl5 30 . . . 4  |-  ( D  e.  V  ->  (
x  e.  dom  N  ->  ( N `  x
)  e.  { 1 ,  -u 1 } ) )
5049ralrimiv 2788 . . 3  |-  ( D  e.  V  ->  A. x  e.  dom  N ( N `
 x )  e. 
{ 1 ,  -u
1 } )
51 ffnfv 5894 . . 3  |-  ( N : dom  N --> { 1 ,  -u 1 }  <->  ( N  Fn  dom  N  /\  A. x  e.  dom  N ( N `  x )  e.  { 1 , 
-u 1 } ) )
5235, 50, 51sylanbrc 646 . 2  |-  ( D  e.  V  ->  N : dom  N --> { 1 ,  -u 1 } )
531, 36, 4psgnvali 27408 . . . . . 6  |-  ( y  e.  dom  N  ->  E. w  e. Word  ran  (pmTrsp `  D ) ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  (
-u 1 ^ ( # `
 w ) ) ) )
5437, 53anim12i 550 . . . . 5  |-  ( ( x  e.  dom  N  /\  y  e.  dom  N )  ->  ( E. z  e. Word  ran  (pmTrsp `  D ) ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) )  /\  E. w  e. Word  ran  (pmTrsp `  D
) ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  ( -u 1 ^ ( # `  w
) ) ) ) )
55 reeanv 2875 . . . . 5  |-  ( E. z  e. Word  ran  (pmTrsp `  D ) E. w  e. Word  ran  (pmTrsp `  D
) ( ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) )  /\  ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  (
-u 1 ^ ( # `
 w ) ) ) )  <->  ( E. z  e. Word  ran  (pmTrsp `  D ) ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) )  /\  E. w  e. Word  ran  (pmTrsp `  D
) ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  ( -u 1 ^ ( # `  w
) ) ) ) )
5654, 55sylibr 204 . . . 4  |-  ( ( x  e.  dom  N  /\  y  e.  dom  N )  ->  E. z  e. Word  ran  (pmTrsp `  D
) E. w  e. Word  ran  (pmTrsp `  D )
( ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  ( -u 1 ^ ( # `  z
) ) )  /\  ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  ( -u 1 ^ ( # `  w
) ) ) ) )
57 ccatcl 11743 . . . . . . . 8  |-  ( ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D
) )  ->  (
z concat  w )  e. Word  ran  (pmTrsp `  D ) )
581, 36, 4psgnvalii 27409 . . . . . . . 8  |-  ( ( D  e.  V  /\  ( z concat  w )  e. Word  ran  (pmTrsp `  D )
)  ->  ( N `  ( S  gsumg  ( z concat  w ) ) )  =  (
-u 1 ^ ( # `
 ( z concat  w
) ) ) )
5957, 58sylan2 461 . . . . . . 7  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( N `  ( S  gsumg  ( z concat  w ) ) )  =  (
-u 1 ^ ( # `
 ( z concat  w
) ) ) )
601symggrp 15103 . . . . . . . . . . 11  |-  ( D  e.  V  ->  S  e.  Grp )
61 grpmnd 14817 . . . . . . . . . . 11  |-  ( S  e.  Grp  ->  S  e.  Mnd )
6260, 61syl 16 . . . . . . . . . 10  |-  ( D  e.  V  ->  S  e.  Mnd )
6336, 1, 2symgtrf 27387 . . . . . . . . . . . 12  |-  ran  (pmTrsp `  D )  C_  ( Base `  S )
64 sswrd 11737 . . . . . . . . . . . 12  |-  ( ran  (pmTrsp `  D )  C_  ( Base `  S
)  -> Word  ran  (pmTrsp `  D )  C_ Word  ( Base `  S ) )
6563, 64ax-mp 8 . . . . . . . . . . 11  |- Word  ran  (pmTrsp `  D )  C_ Word  ( Base `  S )
6665sseli 3344 . . . . . . . . . 10  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  z  e. Word  (
Base `  S )
)
6765sseli 3344 . . . . . . . . . 10  |-  ( w  e. Word  ran  (pmTrsp `  D
)  ->  w  e. Word  (
Base `  S )
)
682, 17gsumccat 14787 . . . . . . . . . 10  |-  ( ( S  e.  Mnd  /\  z  e. Word  ( Base `  S
)  /\  w  e. Word  (
Base `  S )
)  ->  ( S  gsumg  ( z concat  w ) )  =  ( ( S 
gsumg  z ) ( +g  `  S ) ( S 
gsumg  w ) ) )
6962, 66, 67, 68syl3an 1226 . . . . . . . . 9  |-  ( ( D  e.  V  /\  z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D
) )  ->  ( S  gsumg  ( z concat  w ) )  =  ( ( S  gsumg  z ) ( +g  `  S ) ( S 
gsumg  w ) ) )
70693expb 1154 . . . . . . . 8  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( S  gsumg  ( z concat 
w ) )  =  ( ( S  gsumg  z ) ( +g  `  S
) ( S  gsumg  w ) ) )
7170fveq2d 5732 . . . . . . 7  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( N `  ( S  gsumg  ( z concat  w ) ) )  =  ( N `  ( ( S  gsumg  z ) ( +g  `  S ) ( S 
gsumg  w ) ) ) )
72 ccatlen 11744 . . . . . . . . . 10  |-  ( ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D
) )  ->  ( # `
 ( z concat  w
) )  =  ( ( # `  z
)  +  ( # `  w ) ) )
7372adantl 453 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( # `  (
z concat  w ) )  =  ( ( # `  z
)  +  ( # `  w ) ) )
7473oveq2d 6097 . . . . . . . 8  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( -u 1 ^ ( # `  (
z concat  w ) ) )  =  ( -u 1 ^ ( ( # `  z )  +  (
# `  w )
) ) )
75 neg1cn 10067 . . . . . . . . . 10  |-  -u 1  e.  CC
7675a1i 11 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  -u 1  e.  CC )
77 lencl 11735 . . . . . . . . . 10  |-  ( w  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  w
)  e.  NN0 )
7877ad2antll 710 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( # `  w
)  e.  NN0 )
7938ad2antrl 709 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( # `  z
)  e.  NN0 )
8076, 78, 79expaddd 11525 . . . . . . . 8  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( -u 1 ^ ( ( # `  z )  +  (
# `  w )
) )  =  ( ( -u 1 ^ ( # `  z
) )  x.  ( -u 1 ^ ( # `  w ) ) ) )
8174, 80eqtrd 2468 . . . . . . 7  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( -u 1 ^ ( # `  (
z concat  w ) ) )  =  ( ( -u
1 ^ ( # `  z ) )  x.  ( -u 1 ^ ( # `  w
) ) ) )
8259, 71, 813eqtr3d 2476 . . . . . 6  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( N `  ( ( S  gsumg  z ) ( +g  `  S
) ( S  gsumg  w ) ) )  =  ( ( -u 1 ^ ( # `  z
) )  x.  ( -u 1 ^ ( # `  w ) ) ) )
83 oveq12 6090 . . . . . . . . 9  |-  ( ( x  =  ( S 
gsumg  z )  /\  y  =  ( S  gsumg  w ) )  ->  ( x
( +g  `  S ) y )  =  ( ( S  gsumg  z ) ( +g  `  S ) ( S 
gsumg  w ) ) )
8483fveq2d 5732 . . . . . . . 8  |-  ( ( x  =  ( S 
gsumg  z )  /\  y  =  ( S  gsumg  w ) )  ->  ( N `  ( x ( +g  `  S ) y ) )  =  ( N `
 ( ( S 
gsumg  z ) ( +g  `  S ) ( S 
gsumg  w ) ) ) )
85 oveq12 6090 . . . . . . . 8  |-  ( ( ( N `  x
)  =  ( -u
1 ^ ( # `  z ) )  /\  ( N `  y )  =  ( -u 1 ^ ( # `  w
) ) )  -> 
( ( N `  x )  x.  ( N `  y )
)  =  ( (
-u 1 ^ ( # `
 z ) )  x.  ( -u 1 ^ ( # `  w
) ) ) )
8684, 85eqeqan12d 2451 . . . . . . 7  |-  ( ( ( x  =  ( S  gsumg  z )  /\  y  =  ( S  gsumg  w ) )  /\  ( ( N `  x )  =  ( -u 1 ^ ( # `  z
) )  /\  ( N `  y )  =  ( -u 1 ^ ( # `  w
) ) ) )  ->  ( ( N `
 ( x ( +g  `  S ) y ) )  =  ( ( N `  x )  x.  ( N `  y )
)  <->  ( N `  ( ( S  gsumg  z ) ( +g  `  S
) ( S  gsumg  w ) ) )  =  ( ( -u 1 ^ ( # `  z
) )  x.  ( -u 1 ^ ( # `  w ) ) ) ) )
8786an4s 800 . . . . . 6  |-  ( ( ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  ( -u 1 ^ ( # `  z
) ) )  /\  ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  ( -u 1 ^ ( # `  w
) ) ) )  ->  ( ( N `
 ( x ( +g  `  S ) y ) )  =  ( ( N `  x )  x.  ( N `  y )
)  <->  ( N `  ( ( S  gsumg  z ) ( +g  `  S
) ( S  gsumg  w ) ) )  =  ( ( -u 1 ^ ( # `  z
) )  x.  ( -u 1 ^ ( # `  w ) ) ) ) )
8882, 87syl5ibrcom 214 . . . . 5  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( ( ( x  =  ( S 
gsumg  z )  /\  ( N `  x )  =  ( -u 1 ^ ( # `  z
) ) )  /\  ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  ( -u 1 ^ ( # `  w
) ) ) )  ->  ( N `  ( x ( +g  `  S ) y ) )  =  ( ( N `  x )  x.  ( N `  y ) ) ) )
8988rexlimdvva 2837 . . . 4  |-  ( D  e.  V  ->  ( E. z  e. Word  ran  (pmTrsp `  D ) E. w  e. Word  ran  (pmTrsp `  D
) ( ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) )  /\  ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  (
-u 1 ^ ( # `
 w ) ) ) )  ->  ( N `  ( x
( +g  `  S ) y ) )  =  ( ( N `  x )  x.  ( N `  y )
) ) )
9056, 89syl5 30 . . 3  |-  ( D  e.  V  ->  (
( x  e.  dom  N  /\  y  e.  dom  N )  ->  ( N `  ( x ( +g  `  S ) y ) )  =  ( ( N `  x )  x.  ( N `  y ) ) ) )
9190imp 419 . 2  |-  ( ( D  e.  V  /\  ( x  e.  dom  N  /\  y  e.  dom  N ) )  ->  ( N `  ( x
( +g  `  S ) y ) )  =  ( ( N `  x )  x.  ( N `  y )
) )
9212, 14, 19, 25, 28, 30, 52, 91isghmd 15015 1  |-  ( D  e.  V  ->  N  e.  ( F  GrpHom  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320   {cpr 3815    _I cid 4493   dom cdm 4878   ran crn 4879   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Fincfn 7109   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995   -ucneg 9292   NN0cn0 10221   ZZcz 10282   ^cexp 11382   #chash 11618  Word cword 11717   concat cconcat 11718   Basecbs 13469   ↾s cress 13470   +g cplusg 13529    gsumg cgsu 13724   Mndcmnd 14684   Grpcgrp 14685  SubGrpcsubg 14938    GrpHom cghm 15003   SymGrpcsymg 15092  mulGrpcmgp 15648  ℂfldccnfld 16703  pmTrspcpmtr 27361  pmSgncpsgn 27391
This theorem is referenced by:  psgnghm2  27415
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1314  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-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-word 11723  df-concat 11724  df-s1 11725  df-substr 11726  df-splice 11727  df-reverse 11728  df-s2 11812  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-subg 14941  df-ghm 15004  df-gim 15046  df-symg 15093  df-oppg 15142  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-drng 15837  df-cnfld 16704  df-pmtr 27362  df-psgn 27392
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