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Theorem isga 15073
 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 15088). 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 15072 . . 3
21elmpt2cl 6291 . 2
3 fvex 5745 . . . . . . . 8
43a1i 11 . . . . . . 7
5 simplr 733 . . . . . . . . 9
6 id 21 . . . . . . . . . . 11
7 simpl 445 . . . . . . . . . . . . 13
87fveq2d 5735 . . . . . . . . . . . 12
9 isga.1 . . . . . . . . . . . 12
108, 9syl6eqr 2488 . . . . . . . . . . 11
116, 10sylan9eqr 2492 . . . . . . . . . 10
1211, 5xpeq12d 4906 . . . . . . . . 9
135, 12oveq12d 6102 . . . . . . . 8
14 simpll 732 . . . . . . . . . . . . . 14
1514fveq2d 5735 . . . . . . . . . . . . 13
16 isga.3 . . . . . . . . . . . . 13
1715, 16syl6eqr 2488 . . . . . . . . . . . 12
1817oveq1d 6099 . . . . . . . . . . 11
1918eqeq1d 2446 . . . . . . . . . 10
2014fveq2d 5735 . . . . . . . . . . . . . . . 16
21 isga.2 . . . . . . . . . . . . . . . 16
2220, 21syl6eqr 2488 . . . . . . . . . . . . . . 15
2322oveqd 6101 . . . . . . . . . . . . . 14
2423oveq1d 6099 . . . . . . . . . . . . 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 6109 . . . . . . 7
3332rabex 4357 . . . . . 6
3431, 1, 33ovmpt2a 6207 . . . . 5
3534eleq2d 2505 . . . 4
36 oveq 6090 . . . . . . . 8
3736eqeq1d 2446 . . . . . . 7
38 oveq 6090 . . . . . . . . 9
39 oveq 6090 . . . . . . . . . 10
40 oveq 6090 . . . . . . . . . . 11
4140oveq2d 6100 . . . . . . . . . 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 5745 . . . . . . 7
519, 50eqeltri 2508 . . . . . 6
52 xpexg 4992 . . . . . 6
5351, 49, 52sylancr 646 . . . . 5
54 elmapg 7034 . . . . 5
5549, 53, 54syl2anc 644 . . . 4
5655anbi1d 687 . . 3
5748, 56bitrd 246 . 2
582, 57biadan2 625 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  crab 2711  cvv 2958  csb 3253   cxp 4879  wf 5453  cfv 5457  (class class class)co 6084   cmap 7021  cbs 13474   cplusg 13534  c0g 13728  cgrp 14690   cga 15071 This theorem is referenced by:  gagrp  15074  gaset  15075  gagrpid  15076  gaf  15077  gaass  15079  ga0  15080  gaid  15081  subgga  15082  gass  15083  gasubg  15084  lactghmga  15112  sylow1lem2  15238  sylow2blem2  15260  sylow3lem1  15266 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-ga 15072
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