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Mirrors > Home > ILE Home > Th. List > sgrpidmndm | Unicode version |
Description: A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.) |
Ref | Expression |
---|---|
sgrpidmnd.b |
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sgrpidmnd.0 |
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Ref | Expression |
---|---|
sgrpidmndm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp-4r 542 |
. . . . . . . . . 10
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2 | simpllr 534 |
. . . . . . . . . . 11
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3 | 2 | 19.8ad 1591 |
. . . . . . . . . 10
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4 | simplr 528 |
. . . . . . . . . . 11
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5 | sgrpidmnd.b |
. . . . . . . . . . . . . 14
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6 | eqid 2177 |
. . . . . . . . . . . . . 14
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7 | sgrpidmnd.0 |
. . . . . . . . . . . . . 14
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8 | 5, 6, 7 | grpidvalg 12671 |
. . . . . . . . . . . . 13
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9 | 8 | eqeq2d 2189 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 9 | ad4antr 494 |
. . . . . . . . . . 11
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11 | 4, 10 | mpbid 147 |
. . . . . . . . . 10
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12 | 1, 3, 11 | 3jca 1177 |
. . . . . . . . 9
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13 | simpr 110 |
. . . . . . . . 9
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14 | eleq1w 2238 |
. . . . . . . . . . . 12
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15 | oveq1 5875 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | eqeq1d 2186 |
. . . . . . . . . . . . 13
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17 | 16 | ovanraleqv 5892 |
. . . . . . . . . . . 12
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18 | 14, 17 | anbi12d 473 |
. . . . . . . . . . 11
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19 | 18 | iotam 5203 |
. . . . . . . . . 10
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20 | rsp 2524 |
. . . . . . . . . 10
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21 | 19, 20 | simpl2im 386 |
. . . . . . . . 9
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22 | 12, 13, 21 | sylc 62 |
. . . . . . . 8
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23 | 22 | ralrimiva 2550 |
. . . . . . 7
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24 | 23 | exp31 364 |
. . . . . 6
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25 | 24 | exlimdv 1819 |
. . . . 5
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26 | 25 | impd 254 |
. . . 4
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27 | 26 | reximdva 2579 |
. . 3
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28 | 27 | imdistani 445 |
. 2
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29 | 5, 6 | ismnddef 12698 |
. 2
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30 | 28, 29 | sylibr 134 |
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 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
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-csb 3058 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 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mnd 12697 |
This theorem is referenced by: (None) |
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