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Theorem archiabllem1a 28346
Description: Lemma for archiabl 28353: In case an archimedean group  W admits a smallest positive element  U, then any positive element  X of  W can be written as  ( n  .x.  U ) with  n  e.  NN. Since the reciprocal holds for negative elements,  W is then isomorphic to  ZZ. (Contributed by Thierry Arnoux, 12-Apr-2018.)
Hypotheses
Ref Expression
archiabllem.b  |-  B  =  ( Base `  W
)
archiabllem.0  |-  .0.  =  ( 0g `  W )
archiabllem.e  |-  .<_  =  ( le `  W )
archiabllem.t  |-  .<  =  ( lt `  W )
archiabllem.m  |-  .x.  =  (.g
`  W )
archiabllem.g  |-  ( ph  ->  W  e. oGrp )
archiabllem.a  |-  ( ph  ->  W  e. Archi )
archiabllem1.u  |-  ( ph  ->  U  e.  B )
archiabllem1.p  |-  ( ph  ->  .0.  .<  U )
archiabllem1.s  |-  ( (
ph  /\  x  e.  B  /\  .0.  .<  x
)  ->  U  .<_  x )
archiabllem1a.x  |-  ( ph  ->  X  e.  B )
archiabllem1a.c  |-  ( ph  ->  .0.  .<  X )
Assertion
Ref Expression
archiabllem1a  |-  ( ph  ->  E. n  e.  NN  X  =  ( n  .x.  U ) )
Distinct variable groups:    x, n, B    U, n, x    n, W, x    n, X, x    ph, n, x    .x. , n, x    .0. , n, x    .< , n, x   
x,  .<_
Allowed substitution hint:    .<_ ( n)

Proof of Theorem archiabllem1a
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 simplr 760 . . . 4  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  NN0 )
2 nn0p1nn 10909 . . . 4  |-  ( m  e.  NN0  ->  ( m  +  1 )  e.  NN )
31, 2syl 17 . . 3  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  +  1 )  e.  NN )
4 archiabllem1.u . . . . . . . 8  |-  ( ph  ->  U  e.  B )
54ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  U  e.  B )
6 archiabllem.b . . . . . . . 8  |-  B  =  ( Base `  W
)
7 archiabllem.m . . . . . . . 8  |-  .x.  =  (.g
`  W )
86, 7mulg1 16716 . . . . . . 7  |-  ( U  e.  B  ->  (
1  .x.  U )  =  U )
95, 8syl 17 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( 1  .x.  U
)  =  U )
109oveq1d 6320 . . . . 5  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( 1  .x. 
U ) ( +g  `  W ) ( m 
.x.  U ) )  =  ( U ( +g  `  W ) ( m  .x.  U
) ) )
11 archiabllem.g . . . . . . . 8  |-  ( ph  ->  W  e. oGrp )
1211ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  W  e. oGrp )
13 ogrpgrp 28304 . . . . . . 7  |-  ( W  e. oGrp  ->  W  e.  Grp )
1412, 13syl 17 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  W  e.  Grp )
15 1zzd 10968 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
1  e.  ZZ )
161nn0zd 11038 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  ZZ )
17 eqid 2429 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
186, 7, 17mulgdir 16734 . . . . . 6  |-  ( ( W  e.  Grp  /\  ( 1  e.  ZZ  /\  m  e.  ZZ  /\  U  e.  B )
)  ->  ( (
1  +  m ) 
.x.  U )  =  ( ( 1  .x. 
U ) ( +g  `  W ) ( m 
.x.  U ) ) )
1914, 15, 16, 5, 18syl13anc 1266 . . . . 5  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( 1  +  m )  .x.  U
)  =  ( ( 1  .x.  U ) ( +g  `  W
) ( m  .x.  U ) ) )
20 isogrp 28303 . . . . . . . . . 10  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
2120simprbi 465 . . . . . . . . 9  |-  ( W  e. oGrp  ->  W  e. oMnd )
22 omndtos 28306 . . . . . . . . 9  |-  ( W  e. oMnd  ->  W  e. Toset )
23 tospos 28257 . . . . . . . . 9  |-  ( W  e. Toset  ->  W  e.  Poset )
2421, 22, 233syl 18 . . . . . . . 8  |-  ( W  e. oGrp  ->  W  e.  Poset )
2512, 24syl 17 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  W  e.  Poset )
26 archiabllem1a.x . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
2726ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  X  e.  B )
286, 7mulgcl 16726 . . . . . . . . 9  |-  ( ( W  e.  Grp  /\  m  e.  ZZ  /\  U  e.  B )  ->  (
m  .x.  U )  e.  B )
2914, 16, 5, 28syl3anc 1264 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  .x.  U
)  e.  B )
30 eqid 2429 . . . . . . . . 9  |-  ( -g `  W )  =  (
-g `  W )
316, 30grpsubcl 16685 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  ( m  .x.  U )  e.  B )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  e.  B )
3214, 27, 29, 31syl3anc 1264 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  e.  B )
3316peano2zd 11043 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  +  1 )  e.  ZZ )
346, 7mulgcl 16726 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  ( m  +  1
)  e.  ZZ  /\  U  e.  B )  ->  ( ( m  + 
1 )  .x.  U
)  e.  B )
3514, 33, 5, 34syl3anc 1264 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  + 
1 )  .x.  U
)  e.  B )
36 simprr 764 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  X  .<_  ( ( m  +  1 )  .x.  U ) )
37 archiabllem.e . . . . . . . . . 10  |-  .<_  =  ( le `  W )
386, 37, 30ogrpsub 28318 . . . . . . . . 9  |-  ( ( W  e. oGrp  /\  ( X  e.  B  /\  ( ( m  + 
1 )  .x.  U
)  e.  B  /\  ( m  .x.  U )  e.  B )  /\  X  .<_  ( ( m  +  1 )  .x.  U ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) ) 
.<_  ( ( ( m  +  1 )  .x.  U ) ( -g `  W ) ( m 
.x.  U ) ) )
3912, 27, 35, 29, 36, 38syl131anc 1277 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) ) 
.<_  ( ( ( m  +  1 )  .x.  U ) ( -g `  W ) ( m 
.x.  U ) ) )
401nn0cnd 10927 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  CC )
41 1cnd 9658 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
1  e.  CC )
4240, 41pncan2d 9987 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  + 
1 )  -  m
)  =  1 )
4342oveq1d 6320 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( ( m  +  1 )  -  m )  .x.  U
)  =  ( 1 
.x.  U ) )
446, 7, 30mulgsubdir 16740 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  ( ( m  + 
1 )  e.  ZZ  /\  m  e.  ZZ  /\  U  e.  B )
)  ->  ( (
( m  +  1 )  -  m ) 
.x.  U )  =  ( ( ( m  +  1 )  .x.  U ) ( -g `  W ) ( m 
.x.  U ) ) )
4514, 33, 16, 5, 44syl13anc 1266 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( ( m  +  1 )  -  m )  .x.  U
)  =  ( ( ( m  +  1 )  .x.  U ) ( -g `  W
) ( m  .x.  U ) ) )
4643, 45, 93eqtr3d 2478 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( ( m  +  1 )  .x.  U ) ( -g `  W ) ( m 
.x.  U ) )  =  U )
4739, 46breqtrd 4450 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) ) 
.<_  U )
48 archiabllem1.s . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  B  /\  .0.  .<  x
)  ->  U  .<_  x )
49483expia 1207 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .<  x  ->  U  .<_  x ) )
5049ralrimiva 2846 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  B  (  .0.  .<  x  ->  U 
.<_  x ) )
5150ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  A. x  e.  B  (  .0.  .<  x  ->  U 
.<_  x ) )
52 archiabllem.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  W )
536, 52, 30grpsubid 16689 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  ( m  .x.  U )  e.  B )  -> 
( ( m  .x.  U ) ( -g `  W ) ( m 
.x.  U ) )  =  .0.  )
5414, 29, 53syl2anc 665 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  .x.  U ) ( -g `  W ) ( m 
.x.  U ) )  =  .0.  )
55 simprl 762 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  .x.  U
)  .<  X )
56 archiabllem.t . . . . . . . . . . 11  |-  .<  =  ( lt `  W )
576, 56, 30ogrpsublt 28323 . . . . . . . . . 10  |-  ( ( W  e. oGrp  /\  (
( m  .x.  U
)  e.  B  /\  X  e.  B  /\  ( m  .x.  U )  e.  B )  /\  ( m  .x.  U ) 
.<  X )  ->  (
( m  .x.  U
) ( -g `  W
) ( m  .x.  U ) )  .< 
( X ( -g `  W ) ( m 
.x.  U ) ) )
5812, 29, 27, 29, 55, 57syl131anc 1277 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  .x.  U ) ( -g `  W ) ( m 
.x.  U ) ) 
.<  ( X ( -g `  W ) ( m 
.x.  U ) ) )
5954, 58eqbrtrrd 4448 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  .0.  .<  ( X (
-g `  W )
( m  .x.  U
) ) )
60 breq2 4430 . . . . . . . . . 10  |-  ( x  =  ( X (
-g `  W )
( m  .x.  U
) )  ->  (  .0.  .<  x  <->  .0.  .<  ( X ( -g `  W
) ( m  .x.  U ) ) ) )
61 breq2 4430 . . . . . . . . . 10  |-  ( x  =  ( X (
-g `  W )
( m  .x.  U
) )  ->  ( U  .<_  x  <->  U  .<_  ( X ( -g `  W
) ( m  .x.  U ) ) ) )
6260, 61imbi12d 321 . . . . . . . . 9  |-  ( x  =  ( X (
-g `  W )
( m  .x.  U
) )  ->  (
(  .0.  .<  x  ->  U  .<_  x )  <->  (  .0.  .<  ( X
( -g `  W ) ( m  .x.  U
) )  ->  U  .<_  ( X ( -g `  W ) ( m 
.x.  U ) ) ) ) )
6362rspcv 3184 . . . . . . . 8  |-  ( ( X ( -g `  W
) ( m  .x.  U ) )  e.  B  ->  ( A. x  e.  B  (  .0.  .<  x  ->  U  .<_  x )  ->  (  .0.  .<  ( X (
-g `  W )
( m  .x.  U
) )  ->  U  .<_  ( X ( -g `  W ) ( m 
.x.  U ) ) ) ) )
6432, 51, 59, 63syl3c 63 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  U  .<_  ( X (
-g `  W )
( m  .x.  U
) ) )
656, 37posasymb 16149 . . . . . . . 8  |-  ( ( W  e.  Poset  /\  ( X ( -g `  W
) ( m  .x.  U ) )  e.  B  /\  U  e.  B )  ->  (
( ( X (
-g `  W )
( m  .x.  U
) )  .<_  U  /\  U  .<_  ( X (
-g `  W )
( m  .x.  U
) ) )  <->  ( X
( -g `  W ) ( m  .x.  U
) )  =  U ) )
6665biimpa 486 . . . . . . 7  |-  ( ( ( W  e.  Poset  /\  ( X ( -g `  W ) ( m 
.x.  U ) )  e.  B  /\  U  e.  B )  /\  (
( X ( -g `  W ) ( m 
.x.  U ) ) 
.<_  U  /\  U  .<_  ( X ( -g `  W
) ( m  .x.  U ) ) ) )  ->  ( X
( -g `  W ) ( m  .x.  U
) )  =  U )
6725, 32, 5, 47, 64, 66syl32anc 1272 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  =  U )
6867oveq1d 6320 . . . . 5  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( X (
-g `  W )
( m  .x.  U
) ) ( +g  `  W ) ( m 
.x.  U ) )  =  ( U ( +g  `  W ) ( m  .x.  U
) ) )
6910, 19, 683eqtr4rd 2481 . . . 4  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( X (
-g `  W )
( m  .x.  U
) ) ( +g  `  W ) ( m 
.x.  U ) )  =  ( ( 1  +  m )  .x.  U ) )
706, 17, 30grpnpcan 16697 . . . . 5  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  ( m  .x.  U )  e.  B )  -> 
( ( X (
-g `  W )
( m  .x.  U
) ) ( +g  `  W ) ( m 
.x.  U ) )  =  X )
7114, 27, 29, 70syl3anc 1264 . . . 4  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( X (
-g `  W )
( m  .x.  U
) ) ( +g  `  W ) ( m 
.x.  U ) )  =  X )
7241, 40addcomd 9834 . . . . 5  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( 1  +  m
)  =  ( m  +  1 ) )
7372oveq1d 6320 . . . 4  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( 1  +  m )  .x.  U
)  =  ( ( m  +  1 ) 
.x.  U ) )
7469, 71, 733eqtr3d 2478 . . 3  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  X  =  ( (
m  +  1 ) 
.x.  U ) )
75 oveq1 6312 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  U )  =  ( ( m  +  1 )  .x.  U ) )
7675eqeq2d 2443 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  ( X  =  ( n  .x.  U )  <->  X  =  ( ( m  + 
1 )  .x.  U
) ) )
7776rspcev 3188 . . 3  |-  ( ( ( m  +  1 )  e.  NN  /\  X  =  ( (
m  +  1 ) 
.x.  U ) )  ->  E. n  e.  NN  X  =  ( n  .x.  U ) )
783, 74, 77syl2anc 665 . 2  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  E. n  e.  NN  X  =  ( n  .x.  U ) )
79 archiabllem.a . . 3  |-  ( ph  ->  W  e. Archi )
80 archiabllem1.p . . 3  |-  ( ph  ->  .0.  .<  U )
81 archiabllem1a.c . . 3  |-  ( ph  ->  .0.  .<  X )
826, 52, 56, 37, 7, 11, 79, 4, 26, 80, 81archirng 28343 . 2  |-  ( ph  ->  E. m  e.  NN0  ( ( m  .x.  U )  .<  X  /\  X  .<_  ( ( m  +  1 )  .x.  U ) ) )
8378, 82r19.29a 2977 1  |-  ( ph  ->  E. n  e.  NN  X  =  ( n  .x.  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   1c1 9539    + caddc 9541    - cmin 9859   NNcn 10609   NN0cn0 10869   ZZcz 10937   Basecbs 15084   +g cplusg 15152   lecple 15159   0gc0g 15297   Posetcpo 16136   ltcplt 16137  Tosetctos 16230   Grpcgrp 16620   -gcsg 16622  .gcmg 16623  oMndcomnd 28298  oGrpcogrp 28299  Archicarchi 28332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-seq 12211  df-0g 15299  df-preset 16124  df-poset 16142  df-plt 16155  df-toset 16231  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-omnd 28300  df-ogrp 28301  df-inftm 28333  df-archi 28334
This theorem is referenced by:  archiabllem1b  28347
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