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Theorem psgnghm 27540
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 2296 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2296 . . . . . 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 27527 . . . . 5  |-  N  Fn  { x  e.  ( Base `  S )  |  dom  ( x  \  _I  )  e.  Fin }
6 fndm 5359 . . . . 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 3271 . . . 4  |-  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  C_  ( Base `  S )
97, 8eqsstri 3221 . . 3  |-  dom  N  C_  ( Base `  S
)
10 psgnghm.f . . . 4  |-  F  =  ( Ss  dom  N )
1110, 2ressbas2 13215 . . 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 27538 . 2  |-  { 1 ,  -u 1 }  =  ( Base `  U )
15 fvex 5555 . . . 4  |-  ( Base `  F )  e.  _V
1612, 15eqeltri 2366 . . 3  |-  dom  N  e.  _V
17 eqid 2296 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
1810, 17ressplusg 13266 . . 3  |-  ( dom 
N  e.  _V  ->  ( +g  `  S )  =  ( +g  `  F
) )
1916, 18ax-mp 8 . 2  |-  ( +g  `  S )  =  ( +g  `  F )
20 prex 4233 . . 3  |-  { 1 ,  -u 1 }  e.  _V
21 eqid 2296 . . . . 5  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
22 cnfldmul 16401 . . . . 5  |-  x.  =  ( .r ` fld )
2321, 22mgpplusg 15345 . . . 4  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2413, 23ressplusg 13266 . . 3  |-  ( { 1 ,  -u 1 }  e.  _V  ->  x.  =  ( +g  `  U
) )
2520, 24ax-mp 8 . 2  |-  x.  =  ( +g  `  U )
261, 4psgndmsubg 27528 . . 3  |-  ( D  e.  V  ->  dom  N  e.  (SubGrp `  S
) )
2710subggrp 14640 . . 3  |-  ( dom 
N  e.  (SubGrp `  S )  ->  F  e.  Grp )
2826, 27syl 15 . 2  |-  ( D  e.  V  ->  F  e.  Grp )
2913cnmsgngrp 27539 . . 3  |-  U  e. 
Grp
3029a1i 10 . 2  |-  ( D  e.  V  ->  U  e.  Grp )
31 fnfun 5357 . . . . . 6  |-  ( N  Fn  { x  e.  ( Base `  S
)  |  dom  (
x  \  _I  )  e.  Fin }  ->  Fun  N )
325, 31ax-mp 8 . . . . 5  |-  Fun  N
33 funfn 5299 . . . . 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 2296 . . . . . 6  |-  ran  (pmTrsp `  D )  =  ran  (pmTrsp `  D )
371, 36, 4psgnvali 27534 . . . . 5  |-  ( x  e.  dom  N  ->  E. z  e. Word  ran  (pmTrsp `  D ) ( x  =  ( S  gsumg  z )  /\  ( N `  x )  =  (
-u 1 ^ ( # `
 z ) ) ) )
38 lencl 11437 . . . . . . . . . . 11  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  z
)  e.  NN0 )
3938nn0zd 10131 . . . . . . . . . 10  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  ( # `  z
)  e.  ZZ )
40 m1expcl2 11141 . . . . . . . . . . 11  |-  ( (
# `  z )  e.  ZZ  ->  ( -u 1 ^ ( # `  z
) )  e.  { -u 1 ,  1 } )
41 prcom 3718 . . . . . . . . . . 11  |-  { -u
1 ,  1 }  =  { 1 , 
-u 1 }
4240, 41syl6eleq 2386 . . . . . . . . . 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 2365 . . . . . . . 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 2680 . . . . 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 2638 . . 3  |-  ( D  e.  V  ->  A. x  e.  dom  N ( N `
 x )  e. 
{ 1 ,  -u
1 } )
51 ffnfv 5701 . . 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 27534 . . . . . 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 2720 . . . . 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 11445 . . . . . . . 8  |-  ( ( z  e. Word  ran  (pmTrsp `  D )  /\  w  e. Word  ran  (pmTrsp `  D
) )  ->  (
z concat  w )  e. Word  ran  (pmTrsp `  D ) )
581, 36, 4psgnvalii 27535 . . . . . . . 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 14796 . . . . . . . . . . 11  |-  ( D  e.  V  ->  S  e.  Grp )
61 grpmnd 14510 . . . . . . . . . . 11  |-  ( S  e.  Grp  ->  S  e.  Mnd )
6260, 61syl 15 . . . . . . . . . 10  |-  ( D  e.  V  ->  S  e.  Mnd )
6336, 1, 2symgtrf 27513 . . . . . . . . . . . 12  |-  ran  (pmTrsp `  D )  C_  ( Base `  S )
64 sswrd 11439 . . . . . . . . . . . 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 3189 . . . . . . . . . 10  |-  ( z  e. Word  ran  (pmTrsp `  D
)  ->  z  e. Word  (
Base `  S )
)
6765sseli 3189 . . . . . . . . . 10  |-  ( w  e. Word  ran  (pmTrsp `  D
)  ->  w  e. Word  (
Base `  S )
)
682, 17gsumccat 14480 . . . . . . . . . 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 5545 . . . . . . 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 11446 . . . . . . . . . 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 5890 . . . . . . . 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 9829 . . . . . . . . . 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 11437 . . . . . . . . . 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 11263 . . . . . . . 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 2328 . . . . . . 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 2336 . . . . . 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 5883 . . . . . . . . 9  |-  ( ( x  =  ( S 
gsumg  z )  /\  y  =  ( S  gsumg  w ) )  ->  ( x
( +g  `  S ) y )  =  ( ( S  gsumg  z ) ( +g  `  S ) ( S 
gsumg  w ) ) )
8483fveq2d 5545 . . . . . . . 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 5883 . . . . . . . 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 2311 . . . . . . 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 2687 . . . 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 14708 1  |-  ( D  e.  V  ->  N  e.  ( F  GrpHom  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {cpr 3654    _I cid 4320   dom cdm 4705   ran crn 4706   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758   -ucneg 9054   NN0cn0 9981   ZZcz 10040   ^cexp 11120   #chash 11353  Word cword 11419   concat cconcat 11420   Basecbs 13164   ↾s cress 13165   +g cplusg 13224    gsumg cgsu 13417   Mndcmnd 14377   Grpcgrp 14378  SubGrpcsubg 14631    GrpHom cghm 14696   SymGrpcsymg 14785  mulGrpcmgp 15341  ℂfldccnfld 16393  pmTrspcpmtr 27487  pmSgncpsgn 27517
This theorem is referenced by:  psgnghm2  27541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-reverse 11430  df-s2 11514  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-ghm 14697  df-gim 14739  df-symg 14786  df-oppg 14835  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-cnfld 16394  df-pmtr 27488  df-psgn 27518
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