Theorem archiabllem1 29747

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.

Step	Hyp	Ref	Expression
1		archiabllem.g	. . 3 |- ( ph -> W e. oGrp )
2		ogrpgrp 29703	. . 3 |- ( W e. oGrp -> W e. Grp )
3	1, 2	syl 17	. 2 |- ( ph -> W e. Grp )
4		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> m e. ZZ )
5	4	zcnd 11483	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> m e. CC )
6		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> n e. ZZ )
7	6	zcnd 11483	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> n e. CC )
8	5, 7	addcomd 10238	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> ( m + n ) = ( n + m ) )
9	8	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> ( ( m + n ) .x. U ) = ( ( n + m ) .x. U ) )
10	3	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> W e. Grp )
11		archiabllem1.u	. . . . . . . . . . . 12 |- ( ph -> U e. B )
12	11	ad2antrr 762	. . . . . . . . . . 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 2622	. . . . . . . . . . . 12 |- ( +g ` W ) = ( +g ` W )
16	13, 14, 15	mulgdir 17573	. . . . . . . . . . 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 ) ) )
17	10, 4, 6, 12, 16	syl13anc 1328	. . . . . . . . . 10 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> ( ( m + n ) .x. U ) = ( ( m .x. U ) ( +g ` W ) ( n .x. U ) ) )
18	13, 14, 15	mulgdir 17573	. . . . . . . . . . 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 ) ) )
19	10, 6, 4, 12, 18	syl13anc 1328	. . . . . . . . . 10 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> ( ( n + m ) .x. U ) = ( ( n .x. U ) ( +g ` W ) ( m .x. U ) ) )
20	9, 17, 19	3eqtr3d 2664	. . . . . . . . 9 |- ( ( ( ph /\ m e. ZZ ) /\ n e. ZZ ) -> ( ( m .x. U ) ( +g ` W ) ( n .x. U ) ) = ( ( n .x. U ) ( +g ` W ) ( m .x. U ) ) )
21	20	adantllr 755	. . . . . . . 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 ) ) )
22	21	adantlr 751	. . . . . . 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 ) ) )
23	22	adantr 481	. . . . . 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 799	. . . . . . 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 477	. . . . . . 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 ) )
26	24, 25	oveq12d 6668	. . . . . 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 ) ) )
27	25, 24	oveq12d 6668	. . . . . 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 ) ) )
28	23, 26, 27	3eqtr4d 2666	. . . . 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 798	. . . . . 6 |- ( ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ m e. ZZ ) /\ y = ( m .x. U ) ) -> ph )
30		simpr1r 1119	. . . . . . 7 |- ( ( ph /\ ( ( y e. B /\ z e. B ) /\ m e. ZZ /\ y = ( m .x. U ) ) ) -> z e. B )
31	30	3anassrs 1290	. . . . . 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 )
38	13, 32, 33, 34, 14, 1, 35, 11, 36, 37	archiabllem1b 29746	. . . . . 6 |- ( ( ph /\ z e. B ) -> E. n e. ZZ z = ( n .x. U ) )
39	29, 31, 38	syl2anc 693	. . . . 5 |- ( ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ m e. ZZ ) /\ y = ( m .x. U ) ) -> E. n e. ZZ z = ( n .x. U ) )
40	28, 39	r19.29a 3078	. . . 4 |- ( ( ( ( ph /\ ( y e. B /\ z e. B ) ) /\ m e. ZZ ) /\ y = ( m .x. U ) ) -> ( y ( +g ` W ) z ) = ( z ( +g ` W ) y ) )
41	13, 32, 33, 34, 14, 1, 35, 11, 36, 37	archiabllem1b 29746	. . . . 5 |- ( ( ph /\ y e. B ) -> E. m e. ZZ y = ( m .x. U ) )
42	41	adantrr 753	. . . 4 |- ( ( ph /\ ( y e. B /\ z e. B ) ) -> E. m e. ZZ y = ( m .x. U ) )
43	40, 42	r19.29a 3078	. . 3 |- ( ( ph /\ ( y e. B /\ z e. B ) ) -> ( y ( +g ` W ) z ) = ( z ( +g ` W ) y ) )
44	43	ralrimivva 2971	. 2 |- ( ph -> A. y e. B A. z e. B ( y ( +g ` W ) z ) = ( z ( +g ` W ) y ) )
45	13, 15	isabl2 18201	. 2 |- ( W e. Abel <-> ( W e. Grp /\ A. y e. B A. z e. B ( y ( +g ` W ) z ) = ( z ( +g ` W ) y ) ) )
46	3, 44, 45	sylanbrc 698	1 |- ( ph -> W e. Abel )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   + caddc 9939   ZZcz 11377   Basecbs 15857   +g cplusg 15941   lecple 15948   0gc0g 16100   ltcplt 16941   Grpcgrp 17422  ._gcmg 17540   Abelcabl 18194  oGrpcogrp 29698  Archicarchi 29731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-0g 16102  df-preset 16928  df-poset 16946  df-plt 16958  df-toset 17034  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-cmn 18195  df-abl 18196  df-omnd 29699  df-ogrp 29700  df-inftm 29732  df-archi 29733

This theorem is referenced by:  archiabl  29752