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Theorem archiabllem1a 26352
Description: Lemma for archiabl 26359: 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 754 . . . 4  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  NN0 )
2 elnn0nn 10732 . . . . 5  |-  ( m  e.  NN0  <->  ( m  e.  CC  /\  ( m  +  1 )  e.  NN ) )
32simprbi 464 . . . 4  |-  ( m  e.  NN0  ->  ( m  +  1 )  e.  NN )
41, 3syl 16 . . 3  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  +  1 )  e.  NN )
5 archiabllem1.u . . . . . . . 8  |-  ( ph  ->  U  e.  B )
65ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  U  e.  B )
7 archiabllem.b . . . . . . . 8  |-  B  =  ( Base `  W
)
8 archiabllem.m . . . . . . . 8  |-  .x.  =  (.g
`  W )
97, 8mulg1 15752 . . . . . . 7  |-  ( U  e.  B  ->  (
1  .x.  U )  =  U )
106, 9syl 16 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( 1  .x.  U
)  =  U )
1110oveq1d 6214 . . . . 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
) ) )
12 archiabllem.g . . . . . . . 8  |-  ( ph  ->  W  e. oGrp )
1312ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  W  e. oGrp )
14 isogrp 26309 . . . . . . . 8  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
1514simplbi 460 . . . . . . 7  |-  ( W  e. oGrp  ->  W  e.  Grp )
1613, 15syl 16 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  W  e.  Grp )
17 1z 10786 . . . . . . 7  |-  1  e.  ZZ
1817a1i 11 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
1  e.  ZZ )
191nn0zd 10855 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  ZZ )
20 eqid 2454 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
217, 8, 20mulgdir 15770 . . . . . 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 ) ) )
2216, 18, 19, 6, 21syl13anc 1221 . . . . 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 ) ) )
2314simprbi 464 . . . . . . . . 9  |-  ( W  e. oGrp  ->  W  e. oMnd )
24 omndtos 26312 . . . . . . . . 9  |-  ( W  e. oMnd  ->  W  e. Toset )
25 tospos 26263 . . . . . . . . 9  |-  ( W  e. Toset  ->  W  e.  Poset )
2623, 24, 253syl 20 . . . . . . . 8  |-  ( W  e. oGrp  ->  W  e.  Poset )
2713, 26syl 16 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  W  e.  Poset )
28 archiabllem1a.x . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
2928ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  X  e.  B )
307, 8mulgcl 15762 . . . . . . . . 9  |-  ( ( W  e.  Grp  /\  m  e.  ZZ  /\  U  e.  B )  ->  (
m  .x.  U )  e.  B )
3116, 19, 6, 30syl3anc 1219 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  .x.  U
)  e.  B )
32 eqid 2454 . . . . . . . . 9  |-  ( -g `  W )  =  (
-g `  W )
337, 32grpsubcl 15724 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  ( m  .x.  U )  e.  B )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  e.  B )
3416, 29, 31, 33syl3anc 1219 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  e.  B )
3519peano2zd 10860 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  +  1 )  e.  ZZ )
367, 8mulgcl 15762 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  ( m  +  1
)  e.  ZZ  /\  U  e.  B )  ->  ( ( m  + 
1 )  .x.  U
)  e.  B )
3716, 35, 6, 36syl3anc 1219 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  + 
1 )  .x.  U
)  e.  B )
38 simprr 756 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  X  .<_  ( ( m  +  1 )  .x.  U ) )
39 archiabllem.e . . . . . . . . . 10  |-  .<_  =  ( le `  W )
407, 39, 32ogrpsub 26324 . . . . . . . . 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 ) ) )
4113, 29, 37, 31, 38, 40syl131anc 1232 . . . . . . . 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 ) ) )
421nn0cnd 10748 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  CC )
43 ax-1cn 9450 . . . . . . . . . . . 12  |-  1  e.  CC
4443a1i 11 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
1  e.  CC )
4542, 44pncan2d 9831 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  + 
1 )  -  m
)  =  1 )
4645oveq1d 6214 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( ( m  +  1 )  -  m )  .x.  U
)  =  ( 1 
.x.  U ) )
477, 8, 32mulgsubdir 15776 . . . . . . . . . 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 ) ) )
4816, 35, 19, 6, 47syl13anc 1221 . . . . . . . . 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 ) ) )
4946, 48, 103eqtr3d 2503 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( ( m  +  1 )  .x.  U ) ( -g `  W ) ( m 
.x.  U ) )  =  U )
5041, 49breqtrd 4423 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) ) 
.<_  U )
51 archiabllem1.s . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  B  /\  .0.  .<  x
)  ->  U  .<_  x )
52513expa 1188 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  B )  /\  .0.  .<  x )  ->  U  .<_  x )
5352ex 434 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .<  x  ->  U  .<_  x ) )
5453ralrimiva 2829 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  B  (  .0.  .<  x  ->  U 
.<_  x ) )
5554ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  A. x  e.  B  (  .0.  .<  x  ->  U 
.<_  x ) )
56 archiabllem.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  W )
577, 56, 32grpsubid 15728 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  ( m  .x.  U )  e.  B )  -> 
( ( m  .x.  U ) ( -g `  W ) ( m 
.x.  U ) )  =  .0.  )
5816, 31, 57syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  .x.  U ) ( -g `  W ) ( m 
.x.  U ) )  =  .0.  )
59 simprl 755 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  .x.  U
)  .<  X )
60 archiabllem.t . . . . . . . . . . 11  |-  .<  =  ( lt `  W )
617, 60, 32ogrpsublt 26329 . . . . . . . . . 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 ) ) )
6213, 31, 29, 31, 59, 61syl131anc 1232 . . . . . . . . 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 ) ) )
6358, 62eqbrtrrd 4421 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  .0.  .<  ( X (
-g `  W )
( m  .x.  U
) ) )
64 breq2 4403 . . . . . . . . . 10  |-  ( x  =  ( X (
-g `  W )
( m  .x.  U
) )  ->  (  .0.  .<  x  <->  .0.  .<  ( X ( -g `  W
) ( m  .x.  U ) ) ) )
65 breq2 4403 . . . . . . . . . 10  |-  ( x  =  ( X (
-g `  W )
( m  .x.  U
) )  ->  ( U  .<_  x  <->  U  .<_  ( X ( -g `  W
) ( m  .x.  U ) ) ) )
6664, 65imbi12d 320 . . . . . . . . 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 ) ) ) ) )
6766rspcv 3173 . . . . . . . 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 ) ) ) ) )
6834, 55, 63, 67syl3c 61 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  U  .<_  ( X (
-g `  W )
( m  .x.  U
) ) )
697, 39posasymb 15240 . . . . . . . 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 ) )
7069biimpa 484 . . . . . . 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 )
7127, 34, 6, 50, 68, 70syl32anc 1227 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  =  U )
7271oveq1d 6214 . . . . 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
) ) )
7311, 22, 723eqtr4rd 2506 . . . 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 ) )
747, 20, 32grpnpcan 15735 . . . . 5  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  ( m  .x.  U )  e.  B )  -> 
( ( X (
-g `  W )
( m  .x.  U
) ) ( +g  `  W ) ( m 
.x.  U ) )  =  X )
7516, 29, 31, 74syl3anc 1219 . . . 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 )
76 addcom 9665 . . . . . 6  |-  ( ( 1  e.  CC  /\  m  e.  CC )  ->  ( 1  +  m
)  =  ( m  +  1 ) )
7744, 42, 76syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( 1  +  m
)  =  ( m  +  1 ) )
7877oveq1d 6214 . . . 4  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( 1  +  m )  .x.  U
)  =  ( ( m  +  1 ) 
.x.  U ) )
7973, 75, 783eqtr3d 2503 . . 3  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  X  =  ( (
m  +  1 ) 
.x.  U ) )
80 oveq1 6206 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  U )  =  ( ( m  +  1 )  .x.  U ) )
8180eqeq2d 2468 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  ( X  =  ( n  .x.  U )  <->  X  =  ( ( m  + 
1 )  .x.  U
) ) )
8281rspcev 3177 . . 3  |-  ( ( ( m  +  1 )  e.  NN  /\  X  =  ( (
m  +  1 ) 
.x.  U ) )  ->  E. n  e.  NN  X  =  ( n  .x.  U ) )
834, 79, 82syl2anc 661 . 2  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  E. n  e.  NN  X  =  ( n  .x.  U ) )
84 archiabllem.a . . 3  |-  ( ph  ->  W  e. Archi )
85 archiabllem1.p . . 3  |-  ( ph  ->  .0.  .<  U )
86 archiabllem1a.c . . 3  |-  ( ph  ->  .0.  .<  X )
877, 56, 60, 39, 8, 12, 84, 5, 28, 85, 86archirng 26349 . 2  |-  ( ph  ->  E. m  e.  NN0  ( ( m  .x.  U )  .<  X  /\  X  .<_  ( ( m  +  1 )  .x.  U ) ) )
8883, 87r19.29a 2966 1  |-  ( ph  ->  E. n  e.  NN  X  =  ( n  .x.  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2798   E.wrex 2799   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   CCcc 9390   1c1 9393    + caddc 9395    - cmin 9705   NNcn 10432   NN0cn0 10689   ZZcz 10756   Basecbs 14291   +g cplusg 14356   lecple 14363   0gc0g 14496   Posetcpo 15228   ltcplt 15229  Tosetctos 15321   Grpcgrp 15528   -gcsg 15531  .gcmg 15532  oMndcomnd 26304  oGrpcogrp 26305  Archicarchi 26338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-seq 11923  df-0g 14498  df-poset 15234  df-plt 15246  df-toset 15322  df-mnd 15533  df-grp 15663  df-minusg 15664  df-sbg 15665  df-mulg 15666  df-omnd 26306  df-ogrp 26307  df-inftm 26339  df-archi 26340
This theorem is referenced by:  archiabllem1b  26353
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