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Mirrors > Home > MPE Home > Th. List > isga | Structured version Visualization version Unicode version |
Description: The predicate "is a (left) group action." The group is said to act on the base set of the action, which is not assumed to have any special properties. There is a related notion of right group action, but as the Wikipedia article explains, it is not mathematically interesting. The way actions are usually thought of is that each element of is a permutation of the elements of (see gapm 17739). Since group theory was classically about symmetry groups, it is therefore likely that the notion of group action was useful even in early group theory. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
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isga.1 | |
isga.2 | |
isga.3 |
Ref | Expression |
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isga |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ga 17723 | . . 3 | |
2 | 1 | elmpt2cl 6876 | . 2 |
3 | fvexd 6203 | . . . . . . 7 | |
4 | simplr 792 | . . . . . . . . 9 | |
5 | id 22 | . . . . . . . . . . 11 | |
6 | simpl 473 | . . . . . . . . . . . . 13 | |
7 | 6 | fveq2d 6195 | . . . . . . . . . . . 12 |
8 | isga.1 | . . . . . . . . . . . 12 | |
9 | 7, 8 | syl6eqr 2674 | . . . . . . . . . . 11 |
10 | 5, 9 | sylan9eqr 2678 | . . . . . . . . . 10 |
11 | 10, 4 | xpeq12d 5140 | . . . . . . . . 9 |
12 | 4, 11 | oveq12d 6668 | . . . . . . . 8 |
13 | simpll 790 | . . . . . . . . . . . . . 14 | |
14 | 13 | fveq2d 6195 | . . . . . . . . . . . . 13 |
15 | isga.3 | . . . . . . . . . . . . 13 | |
16 | 14, 15 | syl6eqr 2674 | . . . . . . . . . . . 12 |
17 | 16 | oveq1d 6665 | . . . . . . . . . . 11 |
18 | 17 | eqeq1d 2624 | . . . . . . . . . 10 |
19 | 13 | fveq2d 6195 | . . . . . . . . . . . . . . . 16 |
20 | isga.2 | . . . . . . . . . . . . . . . 16 | |
21 | 19, 20 | syl6eqr 2674 | . . . . . . . . . . . . . . 15 |
22 | 21 | oveqd 6667 | . . . . . . . . . . . . . 14 |
23 | 22 | oveq1d 6665 | . . . . . . . . . . . . 13 |
24 | 23 | eqeq1d 2624 | . . . . . . . . . . . 12 |
25 | 10, 24 | raleqbidv 3152 | . . . . . . . . . . 11 |
26 | 10, 25 | raleqbidv 3152 | . . . . . . . . . 10 |
27 | 18, 26 | anbi12d 747 | . . . . . . . . 9 |
28 | 4, 27 | raleqbidv 3152 | . . . . . . . 8 |
29 | 12, 28 | rabeqbidv 3195 | . . . . . . 7 |
30 | 3, 29 | csbied 3560 | . . . . . 6 |
31 | ovex 6678 | . . . . . . 7 | |
32 | 31 | rabex 4813 | . . . . . 6 |
33 | 30, 1, 32 | ovmpt2a 6791 | . . . . 5 |
34 | 33 | eleq2d 2687 | . . . 4 |
35 | oveq 6656 | . . . . . . . 8 | |
36 | 35 | eqeq1d 2624 | . . . . . . 7 |
37 | oveq 6656 | . . . . . . . . 9 | |
38 | oveq 6656 | . . . . . . . . . 10 | |
39 | oveq 6656 | . . . . . . . . . . 11 | |
40 | 39 | oveq2d 6666 | . . . . . . . . . 10 |
41 | 38, 40 | eqtrd 2656 | . . . . . . . . 9 |
42 | 37, 41 | eqeq12d 2637 | . . . . . . . 8 |
43 | 42 | 2ralbidv 2989 | . . . . . . 7 |
44 | 36, 43 | anbi12d 747 | . . . . . 6 |
45 | 44 | ralbidv 2986 | . . . . 5 |
46 | 45 | elrab 3363 | . . . 4 |
47 | 34, 46 | syl6bb 276 | . . 3 |
48 | simpr 477 | . . . . 5 | |
49 | fvex 6201 | . . . . . . 7 | |
50 | 8, 49 | eqeltri 2697 | . . . . . 6 |
51 | xpexg 6960 | . . . . . 6 | |
52 | 50, 48, 51 | sylancr 695 | . . . . 5 |
53 | 48, 52 | elmapd 7871 | . . . 4 |
54 | 53 | anbi1d 741 | . . 3 |
55 | 47, 54 | bitrd 268 | . 2 |
56 | 2, 55 | biadan2 674 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 crab 2916 cvv 3200 csb 3533 cxp 5112 wf 5884 cfv 5888 (class class class)co 6650 cmap 7857 cbs 15857 cplusg 15941 c0g 16100 cgrp 17422 cga 17722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-ga 17723 |
This theorem is referenced by: gagrp 17725 gaset 17726 gagrpid 17727 gaf 17728 gaass 17730 ga0 17731 gaid 17732 subgga 17733 gass 17734 gasubg 17735 lactghmga 17824 sylow1lem2 18014 sylow2blem2 18036 sylow3lem1 18042 |
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