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Mirrors > Home > ILE Home > Th. List > sgrp1 | Unicode version |
Description: The structure with one element and the only closed internal operation for a singleton is a semigroup. (Contributed by AV, 10-Feb-2020.) |
Ref | Expression |
---|---|
sgrp1.m |
Ref | Expression |
---|---|
sgrp1 | Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgrp1.m | . . 3 | |
2 | 1 | mgm1 12653 | . 2 Mgm |
3 | df-ov 5868 | . . . . . . 7 | |
4 | opexg 4222 | . . . . . . . . 9 | |
5 | 4 | anidms 397 | . . . . . . . 8 |
6 | fvsng 5704 | . . . . . . . 8 | |
7 | 5, 6 | mpancom 422 | . . . . . . 7 |
8 | 3, 7 | eqtrid 2220 | . . . . . 6 |
9 | 8 | oveq1d 5880 | . . . . 5 |
10 | 8 | oveq2d 5881 | . . . . 5 |
11 | 9, 10 | eqtr4d 2211 | . . . 4 |
12 | oveq1 5872 | . . . . . . . . 9 | |
13 | 12 | oveq1d 5880 | . . . . . . . 8 |
14 | oveq1 5872 | . . . . . . . 8 | |
15 | 13, 14 | eqeq12d 2190 | . . . . . . 7 |
16 | 15 | 2ralbidv 2499 | . . . . . 6 |
17 | 16 | ralsng 3629 | . . . . 5 |
18 | oveq2 5873 | . . . . . . . . 9 | |
19 | 18 | oveq1d 5880 | . . . . . . . 8 |
20 | oveq1 5872 | . . . . . . . . 9 | |
21 | 20 | oveq2d 5881 | . . . . . . . 8 |
22 | 19, 21 | eqeq12d 2190 | . . . . . . 7 |
23 | 22 | ralbidv 2475 | . . . . . 6 |
24 | 23 | ralsng 3629 | . . . . 5 |
25 | oveq2 5873 | . . . . . . 7 | |
26 | oveq2 5873 | . . . . . . . 8 | |
27 | 26 | oveq2d 5881 | . . . . . . 7 |
28 | 25, 27 | eqeq12d 2190 | . . . . . 6 |
29 | 28 | ralsng 3629 | . . . . 5 |
30 | 17, 24, 29 | 3bitrd 214 | . . . 4 |
31 | 11, 30 | mpbird 167 | . . 3 |
32 | snexg 4179 | . . . . 5 | |
33 | elex 2746 | . . . . . . 7 | |
34 | opexg 4222 | . . . . . . 7 | |
35 | 5, 33, 34 | syl2anc 411 | . . . . . 6 |
36 | snexg 4179 | . . . . . 6 | |
37 | 35, 36 | syl 14 | . . . . 5 |
38 | 1 | grpbaseg 12537 | . . . . 5 |
39 | 32, 37, 38 | syl2anc 411 | . . . 4 |
40 | 1 | grpplusgg 12538 | . . . . . . . . 9 |
41 | 32, 37, 40 | syl2anc 411 | . . . . . . . 8 |
42 | 41 | oveqd 5882 | . . . . . . . 8 |
43 | eqidd 2176 | . . . . . . . 8 | |
44 | 41, 42, 43 | oveq123d 5886 | . . . . . . 7 |
45 | eqidd 2176 | . . . . . . . 8 | |
46 | 41 | oveqd 5882 | . . . . . . . 8 |
47 | 41, 45, 46 | oveq123d 5886 | . . . . . . 7 |
48 | 44, 47 | eqeq12d 2190 | . . . . . 6 |
49 | 39, 48 | raleqbidv 2682 | . . . . 5 |
50 | 39, 49 | raleqbidv 2682 | . . . 4 |
51 | 39, 50 | raleqbidv 2682 | . . 3 |
52 | 31, 51 | mpbid 147 | . 2 |
53 | eqid 2175 | . . 3 | |
54 | eqid 2175 | . . 3 | |
55 | 53, 54 | issgrp 12673 | . 2 Smgrp Mgm |
56 | 2, 52, 55 | sylanbrc 417 | 1 Smgrp |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1353 wcel 2146 wral 2453 cvv 2735 csn 3589 cpr 3590 cop 3592 cfv 5208 (class class class)co 5865 cnx 12424 cbs 12427 cplusg 12491 Mgmcmgm 12637 Smgrpcsgrp 12671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-pre-ltirr 7898 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-iota 5170 df-fun 5210 df-fn 5211 df-fv 5216 df-ov 5868 df-pnf 7968 df-mnf 7969 df-ltxr 7971 df-inn 8891 df-2 8949 df-ndx 12430 df-slot 12431 df-base 12433 df-plusg 12504 df-mgm 12639 df-sgrp 12672 |
This theorem is referenced by: mnd1 12708 |
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