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Theorem archiabllem1 27971
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 ogrpgrp 27927 . . 3  |-  ( W  e. oGrp  ->  W  e.  Grp )
31, 2syl 16 . 2  |-  ( ph  ->  W  e.  Grp )
4 simplr 753 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  m  e.  ZZ )
54zcnd 10966 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  m  e.  CC )
6 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
76zcnd 10966 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  n  e.  CC )
85, 7addcomd 9771 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
m  +  n )  =  ( n  +  m ) )
98oveq1d 6285 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  +  n
)  .x.  U )  =  ( ( n  +  m )  .x.  U ) )
103ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  W  e.  Grp )
11 archiabllem1.u . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  B )
1211ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  U  e.  B )
13 archiabllem.b . . . . . . . . . . . 12  |-  B  =  ( Base `  W
)
14 archiabllem.m . . . . . . . . . . . 12  |-  .x.  =  (.g
`  W )
15 eqid 2454 . . . . . . . . . . . 12  |-  ( +g  `  W )  =  ( +g  `  W )
1613, 14, 15mulgdir 16366 . . . . . . . . . . 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 ) ) )
1710, 4, 6, 12, 16syl13anc 1228 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  +  n
)  .x.  U )  =  ( ( m 
.x.  U ) ( +g  `  W ) ( n  .x.  U
) ) )
1813, 14, 15mulgdir 16366 . . . . . . . . . . 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 ) ) )
1910, 6, 4, 12, 18syl13anc 1228 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( n  +  m
)  .x.  U )  =  ( ( n 
.x.  U ) ( +g  `  W ) ( m  .x.  U
) ) )
209, 17, 193eqtr3d 2503 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  ZZ )  /\  n  e.  ZZ )  ->  (
( m  .x.  U
) ( +g  `  W
) ( n  .x.  U ) )  =  ( ( n  .x.  U ) ( +g  `  W ) ( m 
.x.  U ) ) )
2120adantllr 716 . . . . . . . 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 ) ) )
2221adantlr 712 . . . . . . 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
) ) )
2322adantr 463 . . . . . 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
) ) )
24 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 ) )
25 simpr 459 . . . . . . 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 ) )
2624, 25oveq12d 6288 . . . . . 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
) ) )
2725, 24oveq12d 6288 . . . . . 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
) ) )
2823, 26, 273eqtr4d 2505 . . . . 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 ) )
29 simplll 757 . . . . . 6  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  ->  ph )
30 simpr1r 1052 . . . . . . 7  |-  ( (
ph  /\  ( (
y  e.  B  /\  z  e.  B )  /\  m  e.  ZZ  /\  y  =  ( m 
.x.  U ) ) )  ->  z  e.  B )
31303anassrs 1216 . . . . . 6  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  -> 
z  e.  B )
32 archiabllem.0 . . . . . . 7  |-  .0.  =  ( 0g `  W )
33 archiabllem.e . . . . . . 7  |-  .<_  =  ( le `  W )
34 archiabllem.t . . . . . . 7  |-  .<  =  ( lt `  W )
35 archiabllem.a . . . . . . 7  |-  ( ph  ->  W  e. Archi )
36 archiabllem1.p . . . . . . 7  |-  ( ph  ->  .0.  .<  U )
37 archiabllem1.s . . . . . . 7  |-  ( (
ph  /\  x  e.  B  /\  .0.  .<  x
)  ->  U  .<_  x )
3813, 32, 33, 34, 14, 1, 35, 11, 36, 37archiabllem1b 27970 . . . . . 6  |-  ( (
ph  /\  z  e.  B )  ->  E. n  e.  ZZ  z  =  ( n  .x.  U ) )
3929, 31, 38syl2anc 659 . . . . 5  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  ->  E. n  e.  ZZ  z  =  ( n  .x.  U ) )
4028, 39r19.29a 2996 . . . 4  |-  ( ( ( ( ph  /\  ( y  e.  B  /\  z  e.  B
) )  /\  m  e.  ZZ )  /\  y  =  ( m  .x.  U ) )  -> 
( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4113, 32, 33, 34, 14, 1, 35, 11, 36, 37archiabllem1b 27970 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  E. m  e.  ZZ  y  =  ( m  .x.  U ) )
4241adantrr 714 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  ->  E. m  e.  ZZ  y  =  ( m  .x.  U ) )
4340, 42r19.29a 2996 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4443ralrimivva 2875 . 2  |-  ( ph  ->  A. y  e.  B  A. z  e.  B  ( y ( +g  `  W ) z )  =  ( z ( +g  `  W ) y ) )
4513, 15isabl2 17005 . 2  |-  ( W  e.  Abel  <->  ( W  e. 
Grp  /\  A. y  e.  B  A. z  e.  B  ( y
( +g  `  W ) z )  =  ( z ( +g  `  W
) y ) ) )
463, 44, 45sylanbrc 662 1  |-  ( ph  ->  W  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270    + caddc 9484   ZZcz 10860   Basecbs 14716   +g cplusg 14784   lecple 14791   0gc0g 14929   ltcplt 15769   Grpcgrp 16252  .gcmg 16255   Abelcabl 16998  oGrpcogrp 27922  Archicarchi 27955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12090  df-0g 14931  df-preset 15756  df-poset 15774  df-plt 15787  df-toset 15863  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-cmn 16999  df-abl 17000  df-omnd 27923  df-ogrp 27924  df-inftm 27956  df-archi 27957
This theorem is referenced by:  archiabl  27976
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