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Theorem isga 17640
Description: The predicate "is a (left) group action." The group 𝐺 is said to act on the base set 𝑌 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 𝑔 of 𝐺 is a permutation of the elements of 𝑌 (see gapm 17655). 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 𝑋 = (Base‘𝐺)
isga.2 + = (+g𝐺)
isga.3 0 = (0g𝐺)
Assertion
Ref Expression
isga ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑦,𝑋,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥, ,𝑦,𝑧
Allowed substitution hints:   + (𝑥,𝑦,𝑧)   𝑋(𝑥)   0 (𝑥,𝑦,𝑧)

Proof of Theorem isga
Dummy variables 𝑔 𝑏 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ga 17639 . . 3 GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ (Base‘𝑔) / 𝑏{𝑚 ∈ (𝑠𝑚 (𝑏 × 𝑠)) ∣ ∀𝑥𝑠 (((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
21elmpt2cl 6830 . 2 ( ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
3 fvex 6160 . . . . . . . 8 (Base‘𝑔) ∈ V
43a1i 11 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) ∈ V)
5 simplr 791 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑠 = 𝑌)
6 id 22 . . . . . . . . . . 11 (𝑏 = (Base‘𝑔) → 𝑏 = (Base‘𝑔))
7 simpl 473 . . . . . . . . . . . . 13 ((𝑔 = 𝐺𝑠 = 𝑌) → 𝑔 = 𝐺)
87fveq2d 6154 . . . . . . . . . . . 12 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) = (Base‘𝐺))
9 isga.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
108, 9syl6eqr 2678 . . . . . . . . . . 11 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) = 𝑋)
116, 10sylan9eqr 2682 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑏 = 𝑋)
1211, 5xpeq12d 5105 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑏 × 𝑠) = (𝑋 × 𝑌))
135, 12oveq12d 6623 . . . . . . . 8 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑠𝑚 (𝑏 × 𝑠)) = (𝑌𝑚 (𝑋 × 𝑌)))
14 simpll 789 . . . . . . . . . . . . . 14 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑔 = 𝐺)
1514fveq2d 6154 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g𝑔) = (0g𝐺))
16 isga.3 . . . . . . . . . . . . 13 0 = (0g𝐺)
1715, 16syl6eqr 2678 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g𝑔) = 0 )
1817oveq1d 6620 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((0g𝑔)𝑚𝑥) = ( 0 𝑚𝑥))
1918eqeq1d 2628 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((0g𝑔)𝑚𝑥) = 𝑥 ↔ ( 0 𝑚𝑥) = 𝑥))
2014fveq2d 6154 . . . . . . . . . . . . . . . 16 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g𝑔) = (+g𝐺))
21 isga.2 . . . . . . . . . . . . . . . 16 + = (+g𝐺)
2220, 21syl6eqr 2678 . . . . . . . . . . . . . . 15 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g𝑔) = + )
2322oveqd 6622 . . . . . . . . . . . . . 14 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑦(+g𝑔)𝑧) = (𝑦 + 𝑧))
2423oveq1d 6620 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((𝑦(+g𝑔)𝑧)𝑚𝑥) = ((𝑦 + 𝑧)𝑚𝑥))
2524eqeq1d 2628 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2611, 25raleqbidv 3146 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2711, 26raleqbidv 3146 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2819, 27anbi12d 746 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))))
295, 28raleqbidv 3146 . . . . . . . 8 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑥𝑠 (((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))))
3013, 29rabeqbidv 3186 . . . . . . 7 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → {𝑚 ∈ (𝑠𝑚 (𝑏 × 𝑠)) ∣ ∀𝑥𝑠 (((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} = {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
314, 30csbied 3546 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) / 𝑏{𝑚 ∈ (𝑠𝑚 (𝑏 × 𝑠)) ∣ ∀𝑥𝑠 (((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} = {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
32 ovex 6633 . . . . . . 7 (𝑌𝑚 (𝑋 × 𝑌)) ∈ V
3332rabex 4778 . . . . . 6 {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} ∈ V
3431, 1, 33ovmpt2a 6745 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝐺 GrpAct 𝑌) = {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
3534eleq2d 2689 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ∈ {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))}))
36 oveq 6611 . . . . . . . 8 (𝑚 = → ( 0 𝑚𝑥) = ( 0 𝑥))
3736eqeq1d 2628 . . . . . . 7 (𝑚 = → (( 0 𝑚𝑥) = 𝑥 ↔ ( 0 𝑥) = 𝑥))
38 oveq 6611 . . . . . . . . 9 (𝑚 = → ((𝑦 + 𝑧)𝑚𝑥) = ((𝑦 + 𝑧) 𝑥))
39 oveq 6611 . . . . . . . . . 10 (𝑚 = → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 (𝑧𝑚𝑥)))
40 oveq 6611 . . . . . . . . . . 11 (𝑚 = → (𝑧𝑚𝑥) = (𝑧 𝑥))
4140oveq2d 6621 . . . . . . . . . 10 (𝑚 = → (𝑦 (𝑧𝑚𝑥)) = (𝑦 (𝑧 𝑥)))
4239, 41eqtrd 2660 . . . . . . . . 9 (𝑚 = → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 (𝑧 𝑥)))
4338, 42eqeq12d 2641 . . . . . . . 8 (𝑚 = → (((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
44432ralbidv 2988 . . . . . . 7 (𝑚 = → (∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
4537, 44anbi12d 746 . . . . . 6 (𝑚 = → ((( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
4645ralbidv 2985 . . . . 5 (𝑚 = → (∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
4746elrab 3351 . . . 4 ( ∈ {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} ↔ ( ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
4835, 47syl6bb 276 . . 3 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ( ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
49 simpr 477 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
50 fvex 6160 . . . . . . 7 (Base‘𝐺) ∈ V
519, 50eqeltri 2700 . . . . . 6 𝑋 ∈ V
52 xpexg 6914 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
5351, 49, 52sylancr 694 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
5449, 53elmapd 7817 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝑌𝑚 (𝑋 × 𝑌)) ↔ :(𝑋 × 𝑌)⟶𝑌))
5554anbi1d 740 . . 3 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (( ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))) ↔ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
5648, 55bitrd 268 . 2 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
572, 56biadan2 673 1 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  {crab 2916  Vcvv 3191  csb 3519   × cxp 5077  wf 5846  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  Basecbs 15776  +gcplusg 15857  0gc0g 16016  Grpcgrp 17338   GrpAct cga 17638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-ga 17639
This theorem is referenced by:  gagrp  17641  gaset  17642  gagrpid  17643  gaf  17644  gaass  17646  ga0  17647  gaid  17648  subgga  17649  gass  17650  gasubg  17651  lactghmga  17740  sylow1lem2  17930  sylow2blem2  17952  sylow3lem1  17958
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