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Theorem archiabllem1 27399
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 27354 . . . 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 10963 . . . . . . . . . . . 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 10963 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  n  e.  CC )
96, 8addcomd 9777 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
m  +  n )  =  ( n  +  m ) )
109oveq1d 6297 . . . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( +g  `  W )  =  ( +g  `  W )
1714, 15, 16mulgdir 15967 . . . . . . . . . . 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 1230 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  +  n
)  .x.  U )  =  ( ( m 
.x.  U ) ( +g  `  W ) ( n  .x.  U
) ) )
1914, 15, 16mulgdir 15967 . . . . . . . . . . 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 1230 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( n  +  m
)  .x.  U )  =  ( ( n 
.x.  U ) ( +g  `  W ) ( m  .x.  U
) ) )
2110, 18, 203eqtr3d 2516 . . . . . . . . 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 6300 . . . . . 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 6300 . . . . . 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 2518 . . . . 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 1054 . . . . . . 7  |-  ( (
ph  /\  ( (
y  e.  B  /\  z  e.  B )  /\  m  e.  ZZ  /\  y  =  ( m 
.x.  U ) ) )  ->  z  e.  B )
32313anassrs 1218 . . . . . 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 27398 . . . . . 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 3003 . . . 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 27398 . . . . 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 3003 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4544ralrimivva 2885 . 2  |-  ( ph  ->  A. y  e.  B  A. z  e.  B  ( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4614, 16isabl2 16602 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282    + caddc 9491   ZZcz 10860   Basecbs 14486   +g cplusg 14551   lecple 14558   0gc0g 14691   ltcplt 15424   Grpcgrp 15723  .gcmg 15727   Abelcabl 16595  oMndcomnd 27349  oGrpcogrp 27350  Archicarchi 27383
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 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12072  df-0g 14693  df-poset 15429  df-plt 15441  df-toset 15517  df-mnd 15728  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-cmn 16596  df-abl 16597  df-omnd 27351  df-ogrp 27352  df-inftm 27384  df-archi 27385
This theorem is referenced by:  archiabl  27404
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