Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  archiabllem1 Structured version   Unicode version

Theorem archiabllem1 26215
Description: Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-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 )
Assertion
Ref Expression
archiabllem1  |-  ( ph  ->  W  e.  Abel )
Distinct variable groups:    x, B    x, U    x, W    ph, x    x, 
.x.    x,  .0.    x,  .<    x,  .<_

Proof of Theorem archiabllem1
Dummy variables  m  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 archiabllem.g . . 3  |-  ( ph  ->  W  e. oGrp )
2 isogrp 26170 . . . 4  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
32simplbi 460 . . 3  |-  ( W  e. oGrp  ->  W  e.  Grp )
41, 3syl 16 . 2  |-  ( ph  ->  W  e.  Grp )
5 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  m  e.  ZZ )
65zcnd 10753 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  m  e.  CC )
7 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
87zcnd 10753 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  n  e.  CC )
96, 8addcomd 9576 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
m  +  n )  =  ( n  +  m ) )
109oveq1d 6111 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  +  n
)  .x.  U )  =  ( ( n  +  m )  .x.  U ) )
114ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  W  e.  Grp )
12 archiabllem1.u . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  B )
1312ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  U  e.  B )
14 archiabllem.b . . . . . . . . . . . 12  |-  B  =  ( Base `  W
)
15 archiabllem.m . . . . . . . . . . . 12  |-  .x.  =  (.g
`  W )
16 eqid 2443 . . . . . . . . . . . 12  |-  ( +g  `  W )  =  ( +g  `  W )
1714, 15, 16mulgdir 15657 . . . . . . . . . . 11  |-  ( ( W  e.  Grp  /\  ( m  e.  ZZ  /\  n  e.  ZZ  /\  U  e.  B )
)  ->  ( (
m  +  n ) 
.x.  U )  =  ( ( m  .x.  U ) ( +g  `  W ) ( n 
.x.  U ) ) )
1811, 5, 7, 13, 17syl13anc 1220 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  +  n
)  .x.  U )  =  ( ( m 
.x.  U ) ( +g  `  W ) ( n  .x.  U
) ) )
1914, 15, 16mulgdir 15657 . . . . . . . . . . 11  |-  ( ( W  e.  Grp  /\  ( n  e.  ZZ  /\  m  e.  ZZ  /\  U  e.  B )
)  ->  ( (
n  +  m ) 
.x.  U )  =  ( ( n  .x.  U ) ( +g  `  W ) ( m 
.x.  U ) ) )
2011, 7, 5, 13, 19syl13anc 1220 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( n  +  m
)  .x.  U )  =  ( ( n 
.x.  U ) ( +g  `  W ) ( m  .x.  U
) ) )
2110, 18, 203eqtr3d 2483 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  .x.  U
) ( +g  `  W
) ( n  .x.  U ) )  =  ( ( n  .x.  U ) ( +g  `  W ) ( m 
.x.  U ) ) )
2221adantllr 718 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  .x.  U
) ( +g  `  W
) ( n  .x.  U ) )  =  ( ( n  .x.  U ) ( +g  `  W ) ( m 
.x.  U ) ) )
2322adantlr 714 . . . . . . 7  |-  ( ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  /\  n  e.  ZZ )  ->  ( ( m  .x.  U ) ( +g  `  W ) ( n 
.x.  U ) )  =  ( ( n 
.x.  U ) ( +g  `  W ) ( m  .x.  U
) ) )
2423adantr 465 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  /\  m  e.  ZZ )  /\  y  =  (
m  .x.  U )
)  /\  n  e.  ZZ )  /\  z  =  ( n  .x.  U ) )  -> 
( ( m  .x.  U ) ( +g  `  W ) ( n 
.x.  U ) )  =  ( ( n 
.x.  U ) ( +g  `  W ) ( m  .x.  U
) ) )
25 simpllr 758 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  /\  m  e.  ZZ )  /\  y  =  (
m  .x.  U )
)  /\  n  e.  ZZ )  /\  z  =  ( n  .x.  U ) )  -> 
y  =  ( m 
.x.  U ) )
26 simpr 461 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  /\  m  e.  ZZ )  /\  y  =  (
m  .x.  U )
)  /\  n  e.  ZZ )  /\  z  =  ( n  .x.  U ) )  -> 
z  =  ( n 
.x.  U ) )
2725, 26oveq12d 6114 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  /\  m  e.  ZZ )  /\  y  =  (
m  .x.  U )
)  /\  n  e.  ZZ )  /\  z  =  ( n  .x.  U ) )  -> 
( y ( +g  `  W ) z )  =  ( ( m 
.x.  U ) ( +g  `  W ) ( n  .x.  U
) ) )
2826, 25oveq12d 6114 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  /\  m  e.  ZZ )  /\  y  =  (
m  .x.  U )
)  /\  n  e.  ZZ )  /\  z  =  ( n  .x.  U ) )  -> 
( z ( +g  `  W ) y )  =  ( ( n 
.x.  U ) ( +g  `  W ) ( m  .x.  U
) ) )
2924, 27, 283eqtr4d 2485 . . . . 5  |-  ( ( ( ( ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  /\  m  e.  ZZ )  /\  y  =  (
m  .x.  U )
)  /\  n  e.  ZZ )  /\  z  =  ( n  .x.  U ) )  -> 
( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
30 simplll 757 . . . . . 6  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  ->  ph )
31 simpr1r 1046 . . . . . . 7  |-  ( (
ph  /\  ( (
y  e.  B  /\  z  e.  B )  /\  m  e.  ZZ  /\  y  =  ( m 
.x.  U ) ) )  ->  z  e.  B )
32313anassrs 1209 . . . . . 6  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  -> 
z  e.  B )
33 archiabllem.0 . . . . . . 7  |-  .0.  =  ( 0g `  W )
34 archiabllem.e . . . . . . 7  |-  .<_  =  ( le `  W )
35 archiabllem.t . . . . . . 7  |-  .<  =  ( lt `  W )
36 archiabllem.a . . . . . . 7  |-  ( ph  ->  W  e. Archi )
37 archiabllem1.p . . . . . . 7  |-  ( ph  ->  .0.  .<  U )
38 archiabllem1.s . . . . . . 7  |-  ( (
ph  /\  x  e.  B  /\  .0.  .<  x
)  ->  U  .<_  x )
3914, 33, 34, 35, 15, 1, 36, 12, 37, 38archiabllem1b 26214 . . . . . 6  |-  ( (
ph  /\  z  e.  B )  ->  E. n  e.  ZZ  z  =  ( n  .x.  U ) )
4030, 32, 39syl2anc 661 . . . . 5  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  ->  E. n  e.  ZZ  z  =  ( n  .x.  U ) )
4129, 40r19.29a 2867 . . . 4  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  -> 
( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4214, 33, 34, 35, 15, 1, 36, 12, 37, 38archiabllem1b 26214 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  E. m  e.  ZZ  y  =  ( m  .x.  U ) )
4342adantrr 716 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  ->  E. m  e.  ZZ  y  =  ( m  .x.  U ) )
4441, 43r19.29a 2867 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4544ralrimivva 2813 . 2  |-  ( ph  ->  A. y  e.  B  A. z  e.  B  ( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4614, 16isabl2 16290 . 2  |-  ( W  e.  Abel  <->  ( W  e. 
Grp  /\  A. y  e.  B  A. z  e.  B  ( y
( +g  `  W ) z )  =  ( z ( +g  `  W
) y ) ) )
474, 45, 46sylanbrc 664 1  |-  ( ph  ->  W  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   class class class wbr 4297   ` cfv 5423  (class class class)co 6096    + caddc 9290   ZZcz 10651   Basecbs 14179   +g cplusg 14243   lecple 14250   0gc0g 14383   ltcplt 15116   Grpcgrp 15415  .gcmg 15419   Abelcabel 16283  oMndcomnd 26165  oGrpcogrp 26166  Archicarchi 26199
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-0g 14385  df-poset 15121  df-plt 15133  df-toset 15209  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-cmn 16284  df-abl 16285  df-omnd 26167  df-ogrp 26168  df-inftm 26200  df-archi 26201
This theorem is referenced by:  archiabl  26220
  Copyright terms: Public domain W3C validator