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Mirrors > Home > ILE Home > Th. List > issgrp | Unicode version |
Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
Ref | Expression |
---|---|
issgrp.b | |
issgrp.o |
Ref | Expression |
---|---|
issgrp | Smgrp Mgm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basfn 12484 | . . . . 5 | |
2 | vex 2738 | . . . . 5 | |
3 | funfvex 5524 | . . . . . 6 | |
4 | 3 | funfni 5308 | . . . . 5 |
5 | 1, 2, 4 | mp2an 426 | . . . 4 |
6 | 5 | a1i 9 | . . 3 |
7 | fveq2 5507 | . . . 4 | |
8 | issgrp.b | . . . 4 | |
9 | 7, 8 | eqtr4di 2226 | . . 3 |
10 | plusgslid 12524 | . . . . . . 7 Slot | |
11 | 10 | slotex 12454 | . . . . . 6 |
12 | 11 | elv 2739 | . . . . 5 |
13 | 12 | a1i 9 | . . . 4 |
14 | fveq2 5507 | . . . . . 6 | |
15 | 14 | adantr 276 | . . . . 5 |
16 | issgrp.o | . . . . 5 | |
17 | 15, 16 | eqtr4di 2226 | . . . 4 |
18 | simplr 528 | . . . . 5 | |
19 | id 19 | . . . . . . . . . 10 | |
20 | oveq 5871 | . . . . . . . . . 10 | |
21 | eqidd 2176 | . . . . . . . . . 10 | |
22 | 19, 20, 21 | oveq123d 5886 | . . . . . . . . 9 |
23 | eqidd 2176 | . . . . . . . . . 10 | |
24 | oveq 5871 | . . . . . . . . . 10 | |
25 | 19, 23, 24 | oveq123d 5886 | . . . . . . . . 9 |
26 | 22, 25 | eqeq12d 2190 | . . . . . . . 8 |
27 | 26 | adantl 277 | . . . . . . 7 |
28 | 18, 27 | raleqbidv 2682 | . . . . . 6 |
29 | 18, 28 | raleqbidv 2682 | . . . . 5 |
30 | 18, 29 | raleqbidv 2682 | . . . 4 |
31 | 13, 17, 30 | sbcied2 2998 | . . 3 |
32 | 6, 9, 31 | sbcied2 2998 | . 2 |
33 | df-sgrp 12672 | . 2 Smgrp Mgm | |
34 | 32, 33 | elrab2 2894 | 1 Smgrp Mgm |
Colors of variables: wff set class |
Syntax hints: wa 104 wb 105 wceq 1353 wcel 2146 wral 2453 cvv 2735 wsbc 2960 wfn 5203 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-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-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 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: issgrpv 12674 issgrpn0 12675 isnsgrp 12676 sgrpmgm 12677 sgrpass 12678 sgrp0 12679 sgrp1 12680 |
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