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Theorem isga 19072
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 19087). 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 19071 . . 3 GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ (Base‘𝑔) / 𝑏{𝑚 ∈ (𝑠m (𝑏 × 𝑠)) ∣ ∀𝑥𝑠 (((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
21elmpocl 7596 . 2 ( ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
3 fvexd 6858 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) ∈ V)
4 simplr 768 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑠 = 𝑌)
5 id 22 . . . . . . . . . . 11 (𝑏 = (Base‘𝑔) → 𝑏 = (Base‘𝑔))
6 simpl 484 . . . . . . . . . . . . 13 ((𝑔 = 𝐺𝑠 = 𝑌) → 𝑔 = 𝐺)
76fveq2d 6847 . . . . . . . . . . . 12 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) = (Base‘𝐺))
8 isga.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
97, 8eqtr4di 2795 . . . . . . . . . . 11 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) = 𝑋)
105, 9sylan9eqr 2799 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑏 = 𝑋)
1110, 4xpeq12d 5665 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑏 × 𝑠) = (𝑋 × 𝑌))
124, 11oveq12d 7376 . . . . . . . 8 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑠m (𝑏 × 𝑠)) = (𝑌m (𝑋 × 𝑌)))
13 simpll 766 . . . . . . . . . . . . . 14 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑔 = 𝐺)
1413fveq2d 6847 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g𝑔) = (0g𝐺))
15 isga.3 . . . . . . . . . . . . 13 0 = (0g𝐺)
1614, 15eqtr4di 2795 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g𝑔) = 0 )
1716oveq1d 7373 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((0g𝑔)𝑚𝑥) = ( 0 𝑚𝑥))
1817eqeq1d 2739 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((0g𝑔)𝑚𝑥) = 𝑥 ↔ ( 0 𝑚𝑥) = 𝑥))
1913fveq2d 6847 . . . . . . . . . . . . . . . 16 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g𝑔) = (+g𝐺))
20 isga.2 . . . . . . . . . . . . . . . 16 + = (+g𝐺)
2119, 20eqtr4di 2795 . . . . . . . . . . . . . . 15 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g𝑔) = + )
2221oveqd 7375 . . . . . . . . . . . . . 14 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑦(+g𝑔)𝑧) = (𝑦 + 𝑧))
2322oveq1d 7373 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((𝑦(+g𝑔)𝑧)𝑚𝑥) = ((𝑦 + 𝑧)𝑚𝑥))
2423eqeq1d 2739 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2510, 24raleqbidv 3320 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2610, 25raleqbidv 3320 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2718, 26anbi12d 632 . . . . . . . . 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)))))
284, 27raleqbidv 3320 . . . . . . . 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)))))
2912, 28rabeqbidv 3425 . . . . . . 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)))})
303, 29csbied 3894 . . . . . 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)))})
31 ovex 7391 . . . . . . 7 (𝑌m (𝑋 × 𝑌)) ∈ V
3231rabex 5290 . . . . . 6 {𝑚 ∈ (𝑌m (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} ∈ V
3330, 1, 32ovmpoa 7511 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝐺 GrpAct 𝑌) = {𝑚 ∈ (𝑌m (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
3433eleq2d 2824 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ∈ {𝑚 ∈ (𝑌m (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))}))
35 oveq 7364 . . . . . . . 8 (𝑚 = → ( 0 𝑚𝑥) = ( 0 𝑥))
3635eqeq1d 2739 . . . . . . 7 (𝑚 = → (( 0 𝑚𝑥) = 𝑥 ↔ ( 0 𝑥) = 𝑥))
37 oveq 7364 . . . . . . . . 9 (𝑚 = → ((𝑦 + 𝑧)𝑚𝑥) = ((𝑦 + 𝑧) 𝑥))
38 oveq 7364 . . . . . . . . . 10 (𝑚 = → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 (𝑧𝑚𝑥)))
39 oveq 7364 . . . . . . . . . . 11 (𝑚 = → (𝑧𝑚𝑥) = (𝑧 𝑥))
4039oveq2d 7374 . . . . . . . . . 10 (𝑚 = → (𝑦 (𝑧𝑚𝑥)) = (𝑦 (𝑧 𝑥)))
4138, 40eqtrd 2777 . . . . . . . . 9 (𝑚 = → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 (𝑧 𝑥)))
4237, 41eqeq12d 2753 . . . . . . . 8 (𝑚 = → (((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
43422ralbidv 3213 . . . . . . 7 (𝑚 = → (∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
4436, 43anbi12d 632 . . . . . 6 (𝑚 = → ((( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
4544ralbidv 3175 . . . . 5 (𝑚 = → (∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
4645elrab 3646 . . . 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)))))
4734, 46bitrdi 287 . . 3 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ( ∈ (𝑌m (𝑋 × 𝑌)) ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
48 simpr 486 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
498fvexi 6857 . . . . . 6 𝑋 ∈ V
50 xpexg 7685 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
5149, 48, 50sylancr 588 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
5248, 51elmapd 8780 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝑌m (𝑋 × 𝑌)) ↔ :(𝑋 × 𝑌)⟶𝑌))
5352anbi1d 631 . . 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))))))
5447, 53bitrd 279 . 2 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
552, 54biadanii 821 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 205  wa 397   = wceq 1542  wcel 2107  wral 3065  {crab 3408  Vcvv 3446  csb 3856   × cxp 5632  wf 6493  cfv 6497  (class class class)co 7358  m cmap 8766  Basecbs 17084  +gcplusg 17134  0gc0g 17322  Grpcgrp 18749   GrpAct cga 19070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-ga 19071
This theorem is referenced by:  gagrp  19073  gaset  19074  gagrpid  19075  gaf  19076  gaass  19078  ga0  19079  gaid  19080  subgga  19081  gass  19082  gasubg  19083  lactghmga  19188  sylow1lem2  19382  sylow2blem2  19404  sylow3lem1  19410
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