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Theorem psgnghm 27437
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 2283 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2283 . . . . . 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 27424 . . . . 5  |-  N  Fn  { x  e.  ( Base `  S )  |  dom  ( x  \  _I  )  e.  Fin }
6 fndm 5343 . . . . 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 3258 . . . 4  |-  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  C_  ( Base `  S )
97, 8eqsstri 3208 . . 3  |-  dom  N  C_  ( Base `  S
)
10 psgnghm.f . . . 4  |-  F  =  ( Ss  dom  N )
1110, 2ressbas2 13199 . . 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 27435 . 2  |-  { 1 ,  -u 1 }  =  ( Base `  U )
15 fvex 5539 . . . 4  |-  ( Base `  F )  e.  _V
1612, 15eqeltri 2353 . . 3  |-  dom  N  e.  _V
17 eqid 2283 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
1810, 17ressplusg 13250 . . 3  |-  ( dom 
N  e.  _V  ->  ( +g  `  S )  =  ( +g  `  F
) )
1916, 18ax-mp 8 . 2  |-  ( +g  `  S )  =  ( +g  `  F )
20 prex 4217 . . 3  |-  { 1 ,  -u 1 }  e.  _V
21 eqid 2283 . . . . 5  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
22 cnfldmul 16385 . . . . 5  |-  x.  =  ( .r ` fld )
2321, 22mgpplusg 15329 . . . 4  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2413, 23ressplusg 13250 . . 3  |-  ( { 1 ,  -u 1 }  e.  _V  ->  x.  =  ( +g  `  U
) )
2520, 24ax-mp 8 . 2  |-  x.  =  ( +g  `  U )
261, 4psgndmsubg 27425 . . 3  |-  ( D  e.  V  ->  dom  N  e.  (SubGrp `  S
) )
2710subggrp 14624 . . 3  |-  ( dom 
N  e.  (SubGrp `  S )  ->  F  e.  Grp )
2826, 27syl 15 . 2  |-  ( D  e.  V  ->  F  e.  Grp )
2913cnmsgngrp 27436 . . 3  |-  U  e. 
Grp
3029a1i 10 . 2  |-  ( D  e.  V  ->  U  e.  Grp )
31 fnfun 5341 . . . . . 6  |-  ( N  Fn  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  ->  Fun  N )
325, 31ax-mp 8 . . . . 5  |-  Fun  N
33 funfn 5283 . . . . 5  |-  ( Fun 
N  <->  N  Fn  dom  N )
3432, 33mpbi 199 . . . 4  |-  N  Fn  dom  N
3534a1i 10 . . 3  |-  ( D  e.  V  ->  N  Fn  dom  N )
36 eqid 2283 . . . . . 6  |-  ran  (pmTrsp `  D )  =  ran  (pmTrsp `  D )
371, 36, 4psgnvali 27431 . . . . 5  |-  ( x  e.  dom  N  ->  E. z  e. Word  ran  (pmTrsp `  D ) ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) ) )
38 lencl 11421 . . . . . . . . . . 11  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  z
)  e.  NN0 )
3938nn0zd 10115 . . . . . . . . . 10  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  z
)  e.  ZZ )
40 m1expcl2 11125 . . . . . . . . . . 11  |-  ( (
# `  z )  e.  ZZ  ->  ( -u 1 ^ ( # `  z
) )  e.  { -u 1 ,  1 } )
41 prcom 3705 . . . . . . . . . . 11  |-  { -u
1 ,  1 }  =  { 1 , 
-u 1 }
4240, 41syl6eleq 2373 . . . . . . . . . 10  |-  ( (
# `  z )  e.  ZZ  ->  ( -u 1 ^ ( # `  z
) )  e.  {
1 ,  -u 1 } )
4339, 42syl 15 . . . . . . . . 9  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( -u 1 ^ ( # `  z
) )  e.  {
1 ,  -u 1 } )
4443adantl 452 . . . . . . . 8  |-  ( ( D  e.  V  /\  z  e. Word  ran  (pmTrsp `  D ) )  -> 
( -u 1 ^ ( # `
 z ) )  e.  { 1 , 
-u 1 } )
45 eleq1a 2352 . . . . . . . 8  |-  ( (
-u 1 ^ ( # `
 z ) )  e.  { 1 , 
-u 1 }  ->  ( ( N `  x
)  =  ( -u
1 ^ ( # `  z ) )  -> 
( N `  x
)  e.  { 1 ,  -u 1 } ) )
4644, 45syl 15 . . . . . . 7  |-  ( ( D  e.  V  /\  z  e. Word  ran  (pmTrsp `  D ) )  -> 
( ( N `  x )  =  (
-u 1 ^ ( # `
 z ) )  ->  ( N `  x )  e.  {
1 ,  -u 1 } ) )
4746adantld 453 . . . . . 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 2667 . . . . 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 28 . . . 4  |-  ( D  e.  V  ->  (
x  e.  dom  N  ->  ( N `  x
)  e.  { 1 ,  -u 1 } ) )
5049ralrimiv 2625 . . 3  |-  ( D  e.  V  ->  A. x  e.  dom  N ( N `
 x )  e. 
{ 1 ,  -u
1 } )
51 ffnfv 5685 . . 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 645 . 2  |-  ( D  e.  V  ->  N : dom  N --> { 1 ,  -u 1 } )
531, 36, 4psgnvali 27431 . . . . . 6  |-  ( y  e.  dom  N  ->  E. w  e. Word  ran  (pmTrsp `  D ) ( y  =  ( S  gsumg  w )  /\  ( N `  y )  =  (
-u 1 ^ ( # `
 w ) ) ) )
5437, 53anim12i 549 . . . . 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 2707 . . . . 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 203 . . . 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 11429 . . . . . . . 8  |-  ( ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D
) )  ->  (
z concat  w )  e. Word  ran  (pmTrsp `  D ) )
581, 36, 4psgnvalii 27432 . . . . . . . 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 460 . . . . . . 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 14780 . . . . . . . . . . 11  |-  ( D  e.  V  ->  S  e.  Grp )
61 grpmnd 14494 . . . . . . . . . . 11  |-  ( S  e.  Grp  ->  S  e.  Mnd )
6260, 61syl 15 . . . . . . . . . 10  |-  ( D  e.  V  ->  S  e.  Mnd )
6336, 1, 2symgtrf 27410 . . . . . . . . . . . 12  |-  ran  (pmTrsp `  D )  C_  ( Base `  S )
64 sswrd 11423 . . . . . . . . . . . 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 3176 . . . . . . . . . 10  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  z  e. Word  (
Base `  S )
)
6765sseli 3176 . . . . . . . . . 10  |-  ( w  e. Word  ran  (pmTrsp `  D
)  ->  w  e. Word  (
Base `  S )
)
682, 17gsumccat 14464 . . . . . . . . . 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 1224 . . . . . . . . 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 1152 . . . . . . . 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 5529 . . . . . . 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 11430 . . . . . . . . . 10  |-  ( ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D
) )  ->  ( # `
 ( z concat  w
) )  =  ( ( # `  z
)  +  ( # `  w ) ) )
7372adantl 452 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( # `  (
z concat  w ) )  =  ( ( # `  z
)  +  ( # `  w ) ) )
7473oveq2d 5874 . . . . . . . 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 9813 . . . . . . . . . 10  |-  -u 1  e.  CC
7675a1i 10 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  -u 1  e.  CC )
77 lencl 11421 . . . . . . . . . 10  |-  ( w  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  w
)  e.  NN0 )
7877ad2antll 709 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( # `  w
)  e.  NN0 )
7938ad2antrl 708 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D ) ) )  ->  ( # `  z
)  e.  NN0 )
8076, 78, 79expaddd 11247 . . . . . . . 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 2315 . . . . . . 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 2323 . . . . . 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 5867 . . . . . . . . 9  |-  ( ( x  =  ( S 
gsumg  z )  /\  y  =  ( S  gsumg  w ) )  ->  ( x
( +g  `  S ) y )  =  ( ( S  gsumg  z ) ( +g  `  S ) ( S 
gsumg  w ) ) )
8483fveq2d 5529 . . . . . . . 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 5867 . . . . . . . 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 2298 . . . . . . 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 799 . . . . . 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 213 . . . . 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 2674 . . . 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 28 . . 3  |-  ( D  e.  V  ->  (
( x  e.  dom  N  /\  y  e.  dom  N )  ->  ( N `  ( x ( +g  `  S ) y ) )  =  ( ( N `  x )  x.  ( N `  y ) ) ) )
9190imp 418 . 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 14692 1  |-  ( D  e.  V  ->  N  e.  ( F  GrpHom  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {cpr 3641    _I cid 4304   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   NN0cn0 9965   ZZcz 10024   ^cexp 11104   #chash 11337  Word cword 11403   concat cconcat 11404   Basecbs 13148   ↾s cress 13149   +g cplusg 13208    gsumg cgsu 13401   Mndcmnd 14361   Grpcgrp 14362  SubGrpcsubg 14615    GrpHom cghm 14680   SymGrpcsymg 14769  mulGrpcmgp 15325  ℂfldccnfld 16377  pmTrspcpmtr 27384  pmSgncpsgn 27414
This theorem is referenced by:  psgnghm2  27438
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1296  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-reverse 11414  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-ghm 14681  df-gim 14723  df-symg 14770  df-oppg 14819  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-cnfld 16378  df-pmtr 27385  df-psgn 27415
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