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Theorem archiabllem1 26143
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 26098 . . . 4  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
32simplbi 457 . . 3  |-  ( W  e. oGrp  ->  W  e.  Grp )
41, 3syl 16 . 2  |-  ( ph  ->  W  e.  Grp )
5 simplr 749 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  m  e.  ZZ )
65zcnd 10744 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  m  e.  CC )
7 simpr 458 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
87zcnd 10744 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  n  e.  CC )
96, 8addcomd 9567 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
m  +  n )  =  ( n  +  m ) )
109oveq1d 6105 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  +  n
)  .x.  U )  =  ( ( n  +  m )  .x.  U ) )
114ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  W  e.  Grp )
12 archiabllem1.u . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  B )
1312ad2antrr 720 . . . . . . . . . . 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 2441 . . . . . . . . . . . 12  |-  ( +g  `  W )  =  ( +g  `  W )
1714, 15, 16mulgdir 15645 . . . . . . . . . . 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 1215 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  +  n
)  .x.  U )  =  ( ( m 
.x.  U ) ( +g  `  W ) ( n  .x.  U
) ) )
1914, 15, 16mulgdir 15645 . . . . . . . . . . 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 1215 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( n  +  m
)  .x.  U )  =  ( ( n 
.x.  U ) ( +g  `  W ) ( m  .x.  U
) ) )
2110, 18, 203eqtr3d 2481 . . . . . . . . 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 713 . . . . . . . 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 709 . . . . . . 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 462 . . . . . 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 753 . . . . . . 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 458 . . . . . . 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 6108 . . . . . 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 6108 . . . . . 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 2483 . . . . 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 752 . . . . . 6  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  ->  ph )
31 simpr1r 1041 . . . . . . 7  |-  ( (
ph  /\  ( (
y  e.  B  /\  z  e.  B )  /\  m  e.  ZZ  /\  y  =  ( m 
.x.  U ) ) )  ->  z  e.  B )
32313anassrs 1204 . . . . . 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 26142 . . . . . 6  |-  ( (
ph  /\  z  e.  B )  ->  E. n  e.  ZZ  z  =  ( n  .x.  U ) )
4030, 32, 39syl2anc 656 . . . . 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 2860 . . . 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 26142 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  E. m  e.  ZZ  y  =  ( m  .x.  U ) )
4342adantrr 711 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  ->  E. m  e.  ZZ  y  =  ( m  .x.  U ) )
4441, 43r19.29a 2860 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4544ralrimivva 2806 . 2  |-  ( ph  ->  A. y  e.  B  A. z  e.  B  ( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4614, 16isabl2 16278 . 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 659 1  |-  ( ph  ->  W  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   class class class wbr 4289   ` cfv 5415  (class class class)co 6090    + caddc 9281   ZZcz 10642   Basecbs 14170   +g cplusg 14234   lecple 14241   0gc0g 14374   ltcplt 15107   Grpcgrp 15406  .gcmg 15410   Abelcabel 16271  oMndcomnd 26093  oGrpcogrp 26094  Archicarchi 26127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-0g 14376  df-poset 15112  df-plt 15124  df-toset 15200  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-cmn 16272  df-abl 16273  df-omnd 26095  df-ogrp 26096  df-inftm 26128  df-archi 26129
This theorem is referenced by:  archiabl  26148
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