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Theorem archiabllem1a 27425
Description: Lemma for archiabl 27432: 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 10838 . . . . 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 15959 . . . . . . 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 6299 . . . . 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 27382 . . . . . . . 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 10894 . . . . . . 7  |-  1  e.  ZZ
1817a1i 11 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
1  e.  ZZ )
191nn0zd 10964 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  ZZ )
20 eqid 2467 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
217, 8, 20mulgdir 15977 . . . . . 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 1230 . . . . 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 27385 . . . . . . . . 9  |-  ( W  e. oMnd  ->  W  e. Toset )
25 tospos 27336 . . . . . . . . 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 15969 . . . . . . . . 9  |-  ( ( W  e.  Grp  /\  m  e.  ZZ  /\  U  e.  B )  ->  (
m  .x.  U )  e.  B )
3116, 19, 6, 30syl3anc 1228 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  .x.  U
)  e.  B )
32 eqid 2467 . . . . . . . . 9  |-  ( -g `  W )  =  (
-g `  W )
337, 32grpsubcl 15928 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  ( m  .x.  U )  e.  B )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  e.  B )
3416, 29, 31, 33syl3anc 1228 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  e.  B )
3519peano2zd 10969 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( m  +  1 )  e.  ZZ )
367, 8mulgcl 15969 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  ( m  +  1
)  e.  ZZ  /\  U  e.  B )  ->  ( ( m  + 
1 )  .x.  U
)  e.  B )
3716, 35, 6, 36syl3anc 1228 . . . . . . . . 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 27397 . . . . . . . . 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 1241 . . . . . . . 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 10854 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  m  e.  CC )
43 ax-1cn 9550 . . . . . . . . . . . 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 9932 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( ( m  + 
1 )  -  m
)  =  1 )
4645oveq1d 6299 . . . . . . . . 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 15983 . . . . . . . . . 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 1230 . . . . . . . . 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 2516 . . . . . . . 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 4471 . . . . . . 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 1196 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  B )  /\  .0.  .<  x )  ->  U  .<_  x )
5352ex 434 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .<  x  ->  U  .<_  x ) )
5453ralrimiva 2878 . . . . . . . . 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 15932 . . . . . . . . . 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 27402 . . . . . . . . . 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 1241 . . . . . . . . 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 4469 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  .0.  .<  ( X (
-g `  W )
( m  .x.  U
) ) )
64 breq2 4451 . . . . . . . . . 10  |-  ( x  =  ( X (
-g `  W )
( m  .x.  U
) )  ->  (  .0.  .<  x  <->  .0.  .<  ( X ( -g `  W
) ( m  .x.  U ) ) ) )
65 breq2 4451 . . . . . . . . . 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 3210 . . . . . . . 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 15439 . . . . . . . 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 1236 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  -> 
( X ( -g `  W ) ( m 
.x.  U ) )  =  U )
7271oveq1d 6299 . . . . 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 2519 . . . 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 15940 . . . . 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 1228 . . . 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 9765 . . . . . 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 6299 . . . 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 2516 . . 3  |-  ( ( ( ph  /\  m  e.  NN0 )  /\  (
( m  .x.  U
)  .<  X  /\  X  .<_  ( ( m  + 
1 )  .x.  U
) ) )  ->  X  =  ( (
m  +  1 ) 
.x.  U ) )
80 oveq1 6291 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  U )  =  ( ( m  +  1 )  .x.  U ) )
8180eqeq2d 2481 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  ( X  =  ( n  .x.  U )  <->  X  =  ( ( m  + 
1 )  .x.  U
) ) )
8281rspcev 3214 . . 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 27422 . 2  |-  ( ph  ->  E. m  e.  NN0  ( ( m  .x.  U )  .<  X  /\  X  .<_  ( ( m  +  1 )  .x.  U ) ) )
8883, 87r19.29a 3003 1  |-  ( ph  ->  E. n  e.  NN  X  =  ( n  .x.  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   CCcc 9490   1c1 9493    + caddc 9495    - cmin 9805   NNcn 10536   NN0cn0 10795   ZZcz 10864   Basecbs 14490   +g cplusg 14555   lecple 14562   0gc0g 14695   Posetcpo 15427   ltcplt 15428  Tosetctos 15520   Grpcgrp 15727   -gcsg 15730  .gcmg 15731  oMndcomnd 27377  oGrpcogrp 27378  Archicarchi 27411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-seq 12076  df-0g 14697  df-poset 15433  df-plt 15445  df-toset 15521  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-omnd 27379  df-ogrp 27380  df-inftm 27412  df-archi 27413
This theorem is referenced by:  archiabllem1b  27426
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