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Theorem isga 15070
 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 15085). 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.)
Hypotheses
Ref Expression
isga.1
isga.2
isga.3
Assertion
Ref Expression
isga
Distinct variable groups:   ,,,   ,,   ,,,   , ,,
Allowed substitution hints:   (,,)   ()   (,,)

Proof of Theorem isga
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ga 15069 . . 3
21elmpt2cl 6290 . 2
3 fvex 5744 . . . . . . . 8
43a1i 11 . . . . . . 7
5 simplr 733 . . . . . . . . 9
6 id 21 . . . . . . . . . . 11
7 simpl 445 . . . . . . . . . . . . 13
87fveq2d 5734 . . . . . . . . . . . 12
9 isga.1 . . . . . . . . . . . 12
108, 9syl6eqr 2488 . . . . . . . . . . 11
116, 10sylan9eqr 2492 . . . . . . . . . 10
1211, 5xpeq12d 4905 . . . . . . . . 9
135, 12oveq12d 6101 . . . . . . . 8
14 simpll 732 . . . . . . . . . . . . . 14
1514fveq2d 5734 . . . . . . . . . . . . 13
16 isga.3 . . . . . . . . . . . . 13
1715, 16syl6eqr 2488 . . . . . . . . . . . 12
1817oveq1d 6098 . . . . . . . . . . 11
1918eqeq1d 2446 . . . . . . . . . 10
2014fveq2d 5734 . . . . . . . . . . . . . . . 16
21 isga.2 . . . . . . . . . . . . . . . 16
2220, 21syl6eqr 2488 . . . . . . . . . . . . . . 15
2322oveqd 6100 . . . . . . . . . . . . . 14
2423oveq1d 6098 . . . . . . . . . . . . 13
2524eqeq1d 2446 . . . . . . . . . . . 12
2611, 25raleqbidv 2918 . . . . . . . . . . 11
2711, 26raleqbidv 2918 . . . . . . . . . 10
2819, 27anbi12d 693 . . . . . . . . 9
295, 28raleqbidv 2918 . . . . . . . 8
3013, 29rabeqbidv 2953 . . . . . . 7
314, 30csbied 3295 . . . . . 6
32 ovex 6108 . . . . . . 7
3332rabex 4356 . . . . . 6
3431, 1, 33ovmpt2a 6206 . . . . 5
3534eleq2d 2505 . . . 4
36 oveq 6089 . . . . . . . 8
3736eqeq1d 2446 . . . . . . 7
38 oveq 6089 . . . . . . . . 9
39 oveq 6089 . . . . . . . . . 10
40 oveq 6089 . . . . . . . . . . 11
4140oveq2d 6099 . . . . . . . . . 10
4239, 41eqtrd 2470 . . . . . . . . 9
4338, 42eqeq12d 2452 . . . . . . . 8
44432ralbidv 2749 . . . . . . 7
4537, 44anbi12d 693 . . . . . 6
4645ralbidv 2727 . . . . 5
4746elrab 3094 . . . 4
4835, 47syl6bb 254 . . 3
49 simpr 449 . . . . 5
50 fvex 5744 . . . . . . 7
519, 50eqeltri 2508 . . . . . 6
52 xpexg 4991 . . . . . 6
5351, 49, 52sylancr 646 . . . . 5
54 elmapg 7033 . . . . 5
5549, 53, 54syl2anc 644 . . . 4
5655anbi1d 687 . . 3
5748, 56bitrd 246 . 2