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Theorem archiabllem1a 26159
Description: Lemma for archiabl 26166: 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 10614 . . . . 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 15625 . . . . . . 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 6101 . . . . 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 26116 . . . . . . . 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 10668 . . . . . . 7  |-  1  e.  ZZ
1817a1i 11 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
1  e.  ZZ )
191nn0zd 10737 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  ZZ )
20 eqid 2438 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
217, 8, 20mulgdir 15643 . . . . . 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 1220 . . . . 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 26119 . . . . . . . . 9  |-  ( W  e. oMnd  ->  W  e. Toset )
25 tospos 26070 . . . . . . . . 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 15635 . . . . . . . . 9  |-  ( ( W  e.  Grp  /\  m  e.  ZZ  /\  U  e.  B )  ->  (
m  .x.  U )  e.  B )
3116, 19, 6, 30syl3anc 1218 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  .x.  U
)  e.  B )
32 eqid 2438 . . . . . . . . 9  |-  ( -g `  W )  =  (
-g `  W )
337, 32grpsubcl 15597 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  ( m  .x.  U )  e.  B )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  e.  B )
3416, 29, 31, 33syl3anc 1218 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  e.  B )
3519peano2zd 10742 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  +  1 )  e.  ZZ )
367, 8mulgcl 15635 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  ( m  +  1
)  e.  ZZ  /\  U  e.  B )  ->  ( ( m  + 
1 )  .x.  U
)  e.  B )
3716, 35, 6, 36syl3anc 1218 . . . . . . . . 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 26131 . . . . . . . . 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 1231 . . . . . . . 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 10630 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  CC )
43 ax-1cn 9332 . . . . . . . . . . . 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 9713 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  + 
1 )  -  m
)  =  1 )
4645oveq1d 6101 . . . . . . . . 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 15649 . . . . . . . . . 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 1220 . . . . . . . . 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 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 )
5041, 49breqtrd 4311 . . . . . . 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 1187 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  B )  /\  .0.  .<  x )  ->  U  .<_  x )
5352ex 434 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .<  x  ->  U  .<_  x ) )
5453ralrimiva 2794 . . . . . . . . 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 15601 . . . . . . . . . 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 26136 . . . . . . . . . 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 1231 . . . . . . . . 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 4309 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  .0.  .<  ( X (
-g `  W )
( m  .x.  U
) ) )
64 breq2 4291 . . . . . . . . . 10  |-  ( x  =  ( X (
-g `  W )
( m  .x.  U
) )  ->  (  .0.  .<  x  <->  .0.  .<  ( X ( -g `  W
) ( m  .x.  U ) ) ) )
65 breq2 4291 . . . . . . . . . 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 3064 . . . . . . . 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 15114 . . . . . . . 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 1226 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  =  U )
7271oveq1d 6101 . . . . 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 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 ) )
747, 20, 32grpnpcan 15608 . . . . 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 1218 . . . 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 9547 . . . . . 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 6101 . . . 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 2478 . . 3  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  X  =  ( (
m  +  1 ) 
.x.  U ) )
80 oveq1 6093 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  U )  =  ( ( m  +  1 )  .x.  U ) )
8180eqeq2d 2449 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  ( X  =  ( n  .x.  U )  <->  X  =  ( ( m  + 
1 )  .x.  U
) ) )
8281rspcev 3068 . . 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 26156 . 2  |-  ( ph  ->  E. m  e.  NN0  ( ( m  .x.  U )  .<  X  /\  X  .<_  ( ( m  +  1 )  .x.  U ) ) )
8883, 87r19.29a 2857 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 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   1c1 9275    + caddc 9277    - cmin 9587   NNcn 10314   NN0cn0 10571   ZZcz 10638   Basecbs 14166   +g cplusg 14230   lecple 14237   0gc0g 14370   Posetcpo 15102   ltcplt 15103  Tosetctos 15195   Grpcgrp 15402   -gcsg 15405  .gcmg 15406  oMndcomnd 26111  oGrpcogrp 26112  Archicarchi 26145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-seq 11799  df-0g 14372  df-poset 15108  df-plt 15120  df-toset 15196  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-omnd 26113  df-ogrp 26114  df-inftm 26146  df-archi 26147
This theorem is referenced by:  archiabllem1b  26160
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