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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 |
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issgrp.o |
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Ref | Expression |
---|---|
issgrp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basfn 12502 |
. . . . 5
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2 | vex 2740 |
. . . . 5
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3 | funfvex 5528 |
. . . . . 6
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4 | 3 | funfni 5312 |
. . . . 5
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5 | 1, 2, 4 | mp2an 426 |
. . . 4
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6 | 5 | a1i 9 |
. . 3
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7 | fveq2 5511 |
. . . 4
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8 | issgrp.b |
. . . 4
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9 | 7, 8 | eqtr4di 2228 |
. . 3
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10 | plusgslid 12551 |
. . . . . . 7
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11 | 10 | slotex 12472 |
. . . . . 6
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12 | 11 | elv 2741 |
. . . . 5
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13 | 12 | a1i 9 |
. . . 4
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14 | fveq2 5511 |
. . . . . 6
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15 | 14 | adantr 276 |
. . . . 5
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16 | issgrp.o |
. . . . 5
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17 | 15, 16 | eqtr4di 2228 |
. . . 4
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18 | simplr 528 |
. . . . 5
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19 | id 19 |
. . . . . . . . . 10
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20 | oveq 5875 |
. . . . . . . . . 10
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21 | eqidd 2178 |
. . . . . . . . . 10
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22 | 19, 20, 21 | oveq123d 5890 |
. . . . . . . . 9
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23 | eqidd 2178 |
. . . . . . . . . 10
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24 | oveq 5875 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 19, 23, 24 | oveq123d 5890 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 22, 25 | eqeq12d 2192 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | adantl 277 |
. . . . . . 7
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28 | 18, 27 | raleqbidv 2684 |
. . . . . 6
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29 | 18, 28 | raleqbidv 2684 |
. . . . 5
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30 | 18, 29 | raleqbidv 2684 |
. . . 4
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31 | 13, 17, 30 | sbcied2 3000 |
. . 3
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32 | 6, 9, 31 | sbcied2 3000 |
. 2
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33 | df-sgrp 12700 |
. 2
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34 | 32, 33 | elrab2 2896 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-cnex 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-ov 5872 df-inn 8909 df-2 8967 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-sgrp 12700 |
This theorem is referenced by: issgrpv 12702 issgrpn0 12703 isnsgrp 12704 sgrpmgm 12705 sgrpass 12706 sgrp0 12707 sgrp1 12708 |
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