Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > archiabl | Structured version Visualization version Unicode version |
Description: Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.) |
Ref | Expression |
---|---|
archiabl | oGrp oppg oGrp Archi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . 5 | |
2 | eqid 2622 | . . . . 5 | |
3 | eqid 2622 | . . . . 5 | |
4 | eqid 2622 | . . . . 5 | |
5 | eqid 2622 | . . . . 5 .g .g | |
6 | simpll1 1100 | . . . . 5 oGrp oppg oGrp Archi oGrp | |
7 | simpll3 1102 | . . . . 5 oGrp oppg oGrp Archi Archi | |
8 | simplr 792 | . . . . 5 oGrp oppg oGrp Archi | |
9 | simprl 794 | . . . . 5 oGrp oppg oGrp Archi | |
10 | simp2 1062 | . . . . . 6 oGrp oppg oGrp Archi | |
11 | simp1rr 1127 | . . . . . 6 oGrp oppg oGrp Archi | |
12 | simp3 1063 | . . . . . 6 oGrp oppg oGrp Archi | |
13 | breq2 4657 | . . . . . . . 8 | |
14 | breq2 4657 | . . . . . . . 8 | |
15 | 13, 14 | imbi12d 334 | . . . . . . 7 |
16 | 15 | rspcv 3305 | . . . . . 6 |
17 | 10, 11, 12, 16 | syl3c 66 | . . . . 5 oGrp oppg oGrp Archi |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 17 | archiabllem1 29747 | . . . 4 oGrp oppg oGrp Archi |
19 | 18 | adantllr 755 | . . 3 oGrp oppg oGrp Archi |
20 | simpr 477 | . . . 4 oGrp oppg oGrp Archi | |
21 | breq2 4657 | . . . . . 6 | |
22 | breq1 4656 | . . . . . . . 8 | |
23 | 22 | imbi2d 330 | . . . . . . 7 |
24 | 23 | ralbidv 2986 | . . . . . 6 |
25 | 21, 24 | anbi12d 747 | . . . . 5 |
26 | 25 | cbvrexv 3172 | . . . 4 |
27 | 20, 26 | sylib 208 | . . 3 oGrp oppg oGrp Archi |
28 | 19, 27 | r19.29a 3078 | . 2 oGrp oppg oGrp Archi |
29 | simpl1 1064 | . . 3 oGrp oppg oGrp Archi oGrp | |
30 | simpl3 1066 | . . 3 oGrp oppg oGrp Archi Archi | |
31 | eqid 2622 | . . 3 | |
32 | simpl2 1065 | . . 3 oGrp oppg oGrp Archi oppg oGrp | |
33 | simpr 477 | . . . . . . . . . 10 oGrp oppg oGrp Archi | |
34 | ralnex 2992 | . . . . . . . . . 10 | |
35 | 33, 34 | sylibr 224 | . . . . . . . . 9 oGrp oppg oGrp Archi |
36 | rexanali 2998 | . . . . . . . . . . . 12 | |
37 | 36 | imbi2i 326 | . . . . . . . . . . 11 |
38 | imnan 438 | . . . . . . . . . . 11 | |
39 | 37, 38 | bitri 264 | . . . . . . . . . 10 |
40 | 39 | ralbii 2980 | . . . . . . . . 9 |
41 | 35, 40 | sylibr 224 | . . . . . . . 8 oGrp oppg oGrp Archi |
42 | 22 | notbid 308 | . . . . . . . . . . . 12 |
43 | 42 | anbi2d 740 | . . . . . . . . . . 11 |
44 | 43 | rexbidv 3052 | . . . . . . . . . 10 |
45 | 21, 44 | imbi12d 334 | . . . . . . . . 9 |
46 | 45 | cbvralv 3171 | . . . . . . . 8 |
47 | 41, 46 | sylib 208 | . . . . . . 7 oGrp oppg oGrp Archi |
48 | 47 | r19.21bi 2932 | . . . . . 6 oGrp oppg oGrp Archi |
49 | 14 | notbid 308 | . . . . . . . 8 |
50 | 13, 49 | anbi12d 747 | . . . . . . 7 |
51 | 50 | cbvrexv 3172 | . . . . . 6 |
52 | 48, 51 | syl6ib 241 | . . . . 5 oGrp oppg oGrp Archi |
53 | 52 | 3impia 1261 | . . . 4 oGrp oppg oGrp Archi |
54 | simp1l1 1154 | . . . . . 6 oGrp oppg oGrp Archi oGrp | |
55 | isogrp 29702 | . . . . . . 7 oGrp oMnd | |
56 | 55 | simprbi 480 | . . . . . 6 oGrp oMnd |
57 | omndtos 29705 | . . . . . 6 oMnd Toset | |
58 | 54, 56, 57 | 3syl 18 | . . . . 5 oGrp oppg oGrp Archi Toset |
59 | simp2 1062 | . . . . 5 oGrp oppg oGrp Archi | |
60 | 1, 3, 4 | tltnle 29662 | . . . . . . . . . 10 Toset |
61 | 60 | bicomd 213 | . . . . . . . . 9 Toset |
62 | 61 | 3com23 1271 | . . . . . . . 8 Toset |
63 | 62 | 3expa 1265 | . . . . . . 7 Toset |
64 | 63 | anbi2d 740 | . . . . . 6 Toset |
65 | 64 | rexbidva 3049 | . . . . 5 Toset |
66 | 58, 59, 65 | syl2anc 693 | . . . 4 oGrp oppg oGrp Archi |
67 | 53, 66 | mpbid 222 | . . 3 oGrp oppg oGrp Archi |
68 | 1, 2, 3, 4, 5, 29, 30, 31, 32, 67 | archiabllem2 29751 | . 2 oGrp oppg oGrp Archi |
69 | 28, 68 | pm2.61dan 832 | 1 oGrp oppg oGrp Archi |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 class class class wbr 4653 cfv 5888 cbs 15857 cplusg 15941 cple 15948 c0g 16100 cplt 16941 Tosetctos 17033 cgrp 17422 .gcmg 17540 oppgcoppg 17775 cabl 18194 oMndcomnd 29697 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-fal 1489 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-tpos 7352 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-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-ple 15961 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-oppg 17776 df-cmn 18195 df-abl 18196 df-omnd 29699 df-ogrp 29700 df-inftm 29732 df-archi 29733 |
This theorem is referenced by: (None) |
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