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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 12473 | . . . . 5 | |
2 | vex 2733 | . . . . 5 | |
3 | funfvex 5513 | . . . . . 6 | |
4 | 3 | funfni 5298 | . . . . 5 |
5 | 1, 2, 4 | mp2an 424 | . . . 4 |
6 | 5 | a1i 9 | . . 3 |
7 | fveq2 5496 | . . . 4 | |
8 | issgrp.b | . . . 4 | |
9 | 7, 8 | eqtr4di 2221 | . . 3 |
10 | plusgslid 12513 | . . . . . . 7 Slot | |
11 | 10 | slotex 12443 | . . . . . 6 |
12 | 11 | elv 2734 | . . . . 5 |
13 | 12 | a1i 9 | . . . 4 |
14 | fveq2 5496 | . . . . . 6 | |
15 | 14 | adantr 274 | . . . . 5 |
16 | issgrp.o | . . . . 5 | |
17 | 15, 16 | eqtr4di 2221 | . . . 4 |
18 | simplr 525 | . . . . 5 | |
19 | id 19 | . . . . . . . . . 10 | |
20 | oveq 5859 | . . . . . . . . . 10 | |
21 | eqidd 2171 | . . . . . . . . . 10 | |
22 | 19, 20, 21 | oveq123d 5874 | . . . . . . . . 9 |
23 | eqidd 2171 | . . . . . . . . . 10 | |
24 | oveq 5859 | . . . . . . . . . 10 | |
25 | 19, 23, 24 | oveq123d 5874 | . . . . . . . . 9 |
26 | 22, 25 | eqeq12d 2185 | . . . . . . . 8 |
27 | 26 | adantl 275 | . . . . . . 7 |
28 | 18, 27 | raleqbidv 2677 | . . . . . 6 |
29 | 18, 28 | raleqbidv 2677 | . . . . 5 |
30 | 18, 29 | raleqbidv 2677 | . . . 4 |
31 | 13, 17, 30 | sbcied2 2992 | . . 3 |
32 | 6, 9, 31 | sbcied2 2992 | . 2 |
33 | df-sgrp 12643 | . 2 Smgrp Mgm | |
34 | 32, 33 | elrab2 2889 | 1 Smgrp Mgm |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 cvv 2730 wsbc 2955 wfn 5193 cfv 5198 (class class class)co 5853 cbs 12416 cplusg 12480 Mgmcmgm 12608 Smgrpcsgrp 12642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 df-ov 5856 df-inn 8879 df-2 8937 df-ndx 12419 df-slot 12420 df-base 12422 df-plusg 12493 df-sgrp 12643 |
This theorem is referenced by: issgrpv 12645 issgrpn0 12646 isnsgrp 12647 sgrpmgm 12648 sgrpass 12649 sgrp0 12650 sgrp1 12651 |
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