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Theorem psgnghm 27305
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 2404 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2404 . . . . . 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 27292 . . . . 5  |-  N  Fn  { x  e.  ( Base `  S )  |  dom  ( x  \  _I  )  e.  Fin }
6 fndm 5503 . . . . 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 3388 . . . 4  |-  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  C_  ( Base `  S )
97, 8eqsstri 3338 . . 3  |-  dom  N  C_  ( Base `  S
)
10 psgnghm.f . . . 4  |-  F  =  ( Ss  dom  N )
1110, 2ressbas2 13475 . . 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 27303 . 2  |-  { 1 ,  -u 1 }  =  ( Base `  U )
15 fvex 5701 . . . 4  |-  ( Base `  F )  e.  _V
1612, 15eqeltri 2474 . . 3  |-  dom  N  e.  _V
17 eqid 2404 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
1810, 17ressplusg 13526 . . 3  |-  ( dom 
N  e.  _V  ->  ( +g  `  S )  =  ( +g  `  F
) )
1916, 18ax-mp 8 . 2  |-  ( +g  `  S )  =  ( +g  `  F )
20 prex 4366 . . 3  |-  { 1 ,  -u 1 }  e.  _V
21 eqid 2404 . . . . 5  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
22 cnfldmul 16664 . . . . 5  |-  x.  =  ( .r ` fld )
2321, 22mgpplusg 15607 . . . 4  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2413, 23ressplusg 13526 . . 3  |-  ( { 1 ,  -u 1 }  e.  _V  ->  x.  =  ( +g  `  U
) )
2520, 24ax-mp 8 . 2  |-  x.  =  ( +g  `  U )
261, 4psgndmsubg 27293 . . 3  |-  ( D  e.  V  ->  dom  N  e.  (SubGrp `  S
) )
2710subggrp 14902 . . 3  |-  ( dom 
N  e.  (SubGrp `  S )  ->  F  e.  Grp )
2826, 27syl 16 . 2  |-  ( D  e.  V  ->  F  e.  Grp )
2913cnmsgngrp 27304 . . 3  |-  U  e. 
Grp
3029a1i 11 . 2  |-  ( D  e.  V  ->  U  e.  Grp )
31 fnfun 5501 . . . . . 6  |-  ( N  Fn  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  ->  Fun  N )
325, 31ax-mp 8 . . . . 5  |-  Fun  N
33 funfn 5441 . . . . 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 2404 . . . . . 6  |-  ran  (pmTrsp `  D )  =  ran  (pmTrsp `  D )
371, 36, 4psgnvali 27299 . . . . 5  |-  ( x  e.  dom  N  ->  E. z  e. Word  ran  (pmTrsp `  D ) ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) ) )
38 lencl 11690 . . . . . . . . . . 11  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  z
)  e.  NN0 )
3938nn0zd 10329 . . . . . . . . . 10  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  z
)  e.  ZZ )
40 m1expcl2 11358 . . . . . . . . . . 11  |-  ( (
# `  z )  e.  ZZ  ->  ( -u 1 ^ ( # `  z
) )  e.  { -u 1 ,  1 } )
41 prcom 3842 . . . . . . . . . . 11  |-  { -u
1 ,  1 }  =  { 1 , 
-u 1 }
4240, 41syl6eleq 2494 . . . . . . . . . 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 2473 . . . . . . . 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 2790 . . . . 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 2748 . . 3  |-  ( D  e.  V  ->  A. x  e.  dom  N ( N `
 x )  e. 
{ 1 ,  -u
1 } )
51 ffnfv 5853 . . 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 27299 . . . . . 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 2835 . . . . 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 11698 . . . . . . . 8  |-  ( ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D
) )  ->  (
z concat  w )  e. Word  ran  (pmTrsp `  D ) )
581, 36, 4psgnvalii 27300 . . . . . . . 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 15058 . . . . . . . . . . 11  |-  ( D  e.  V  ->  S  e.  Grp )
61 grpmnd 14772 . . . . . . . . . . 11  |-  ( S  e.  Grp  ->  S  e.  Mnd )
6260, 61syl 16 . . . . . . . . . 10  |-  ( D  e.  V  ->  S  e.  Mnd )
6336, 1, 2symgtrf 27278 . . . . . . . . . . . 12  |-  ran  (pmTrsp `  D )  C_  ( Base `  S )
64 sswrd 11692 . . . . . . . . . . . 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 3304 . . . . . . . . . 10  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  z  e. Word  (
Base `  S )
)
6765sseli 3304 . . . . . . . . . 10  |-  ( w  e. Word  ran  (pmTrsp `  D
)  ->  w  e. Word  (
Base `  S )
)
682, 17gsumccat 14742 . . . . . . . . . 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 5691 . . . . . . 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 11699 . . . . . . . . . 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 6056 . . . . . . . 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 10023 . . . . . . . . . 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 11690 . . . . . . . . . 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 11480 . . . . . . . 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 2436 . . . . . . 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 2444 . . . . . 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 6049 . . . . . . . . 9  |-  ( ( x  =  ( S 
gsumg  z )  /\  y  =  ( S  gsumg  w ) )  ->  ( x
( +g  `  S ) y )  =  ( ( S  gsumg  z ) ( +g  `  S ) ( S 
gsumg  w ) ) )
8483fveq2d 5691 . . . . . . . 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 6049 . . . . . . . 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 2419 . . . . . . 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 2797 . . . 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 14970 1  |-  ( D  e.  V  ->  N  e.  ( F  GrpHom  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {cpr 3775    _I cid 4453   dom cdm 4837   ran crn 4838   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951   -ucneg 9248   NN0cn0 10177   ZZcz 10238   ^cexp 11337   #chash 11573  Word cword 11672   concat cconcat 11673   Basecbs 13424   ↾s cress 13425   +g cplusg 13484    gsumg cgsu 13679   Mndcmnd 14639   Grpcgrp 14640  SubGrpcsubg 14893    GrpHom cghm 14958   SymGrpcsymg 15047  mulGrpcmgp 15603  ℂfldccnfld 16658  pmTrspcpmtr 27252  pmSgncpsgn 27282
This theorem is referenced by:  psgnghm2  27306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-reverse 11683  df-s2 11767  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-subg 14896  df-ghm 14959  df-gim 15001  df-symg 15048  df-oppg 15097  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-cnfld 16659  df-pmtr 27253  df-psgn 27283
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