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Mirrors > Home > ILE Home > Th. List > isnsgrp | Unicode version |
Description: A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.) |
Ref | Expression |
---|---|
issgrpn0.b | |
issgrpn0.o |
Ref | Expression |
---|---|
isnsgrp | Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1000 | . . . . . . 7 | |
2 | oveq1 5872 | . . . . . . . . . . . . 13 | |
3 | 2 | oveq1d 5880 | . . . . . . . . . . . 12 |
4 | oveq1 5872 | . . . . . . . . . . . 12 | |
5 | 3, 4 | eqeq12d 2190 | . . . . . . . . . . 11 |
6 | 5 | notbid 667 | . . . . . . . . . 10 |
7 | 6 | rexbidv 2476 | . . . . . . . . 9 |
8 | 7 | rexbidv 2476 | . . . . . . . 8 |
9 | 8 | adantl 277 | . . . . . . 7 |
10 | simpl2 1001 | . . . . . . . 8 | |
11 | oveq2 5873 | . . . . . . . . . . . . 13 | |
12 | 11 | oveq1d 5880 | . . . . . . . . . . . 12 |
13 | oveq1 5872 | . . . . . . . . . . . . 13 | |
14 | 13 | oveq2d 5881 | . . . . . . . . . . . 12 |
15 | 12, 14 | eqeq12d 2190 | . . . . . . . . . . 11 |
16 | 15 | notbid 667 | . . . . . . . . . 10 |
17 | 16 | adantl 277 | . . . . . . . . 9 |
18 | 17 | rexbidv 2476 | . . . . . . . 8 |
19 | simpl3 1002 | . . . . . . . . 9 | |
20 | oveq2 5873 | . . . . . . . . . . . 12 | |
21 | oveq2 5873 | . . . . . . . . . . . . 13 | |
22 | 21 | oveq2d 5881 | . . . . . . . . . . . 12 |
23 | 20, 22 | eqeq12d 2190 | . . . . . . . . . . 11 |
24 | 23 | notbid 667 | . . . . . . . . . 10 |
25 | 24 | adantl 277 | . . . . . . . . 9 |
26 | neneq 2367 | . . . . . . . . . 10 | |
27 | 26 | adantl 277 | . . . . . . . . 9 |
28 | 19, 25, 27 | rspcedvd 2845 | . . . . . . . 8 |
29 | 10, 18, 28 | rspcedvd 2845 | . . . . . . 7 |
30 | 1, 9, 29 | rspcedvd 2845 | . . . . . 6 |
31 | rexnalim 2464 | . . . . . . . . 9 | |
32 | 31 | reximi 2572 | . . . . . . . 8 |
33 | rexnalim 2464 | . . . . . . . 8 | |
34 | 32, 33 | syl 14 | . . . . . . 7 |
35 | 34 | reximi 2572 | . . . . . 6 |
36 | rexnalim 2464 | . . . . . 6 | |
37 | 30, 35, 36 | 3syl 17 | . . . . 5 |
38 | 37 | intnand 931 | . . . 4 Mgm |
39 | issgrpn0.b | . . . . 5 | |
40 | issgrpn0.o | . . . . 5 | |
41 | 39, 40 | issgrp 12673 | . . . 4 Smgrp Mgm |
42 | 38, 41 | sylnibr 677 | . . 3 Smgrp |
43 | df-nel 2441 | . . 3 Smgrp Smgrp | |
44 | 42, 43 | sylibr 134 | . 2 Smgrp |
45 | 44 | ex 115 | 1 Smgrp |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wb 105 w3a 978 wceq 1353 wcel 2146 wne 2345 wnel 2440 wral 2453 wrex 2454 cfv 5208 (class class class)co 5865 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-cnex 7877 ax-resscn 7878 ax-1re 7880 ax-addrcl 7883 |
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-un 3131 df-in 3133 df-ss 3140 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-inn 8891 df-2 8949 df-ndx 12430 df-slot 12431 df-base 12433 df-plusg 12504 df-sgrp 12672 |
This theorem is referenced by: (None) |
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