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| Mirrors > Home > ILE Home > Th. List > isgrp | Unicode version | ||
| Description: The predicate "is a group". (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) |
| Ref | Expression |
|---|---|
| isgrp.b |
|
| isgrp.p |
|
| isgrp.z |
|
| Ref | Expression |
|---|---|
| isgrp |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 5575 |
. . . 4
| |
| 2 | isgrp.b |
. . . 4
| |
| 3 | 1, 2 | eqtr4di 2255 |
. . 3
|
| 4 | fveq2 5575 |
. . . . . . 7
| |
| 5 | isgrp.p |
. . . . . . 7
| |
| 6 | 4, 5 | eqtr4di 2255 |
. . . . . 6
|
| 7 | 6 | oveqd 5960 |
. . . . 5
|
| 8 | fveq2 5575 |
. . . . . 6
| |
| 9 | isgrp.z |
. . . . . 6
| |
| 10 | 8, 9 | eqtr4di 2255 |
. . . . 5
|
| 11 | 7, 10 | eqeq12d 2219 |
. . . 4
|
| 12 | 3, 11 | rexeqbidv 2718 |
. . 3
|
| 13 | 3, 12 | raleqbidv 2717 |
. 2
|
| 14 | df-grp 13306 |
. 2
| |
| 15 | 13, 14 | elrab2 2931 |
1
|
| 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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-un 3169 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-iota 5231 df-fv 5278 df-ov 5946 df-grp 13306 |
| This theorem is referenced by: grpmnd 13310 grpinvex 13313 grppropd 13320 isgrpd2e 13323 grp1 13409 ghmgrp 13425 |
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