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Theorem isga 17924
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 17939). 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 17923 . . 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 7041 . 2 ( ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
3 fvexd 6364 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) ∈ V)
4 simplr 809 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑠 = 𝑌)
5 id 22 . . . . . . . . . . 11 (𝑏 = (Base‘𝑔) → 𝑏 = (Base‘𝑔))
6 simpl 474 . . . . . . . . . . . . 13 ((𝑔 = 𝐺𝑠 = 𝑌) → 𝑔 = 𝐺)
76fveq2d 6356 . . . . . . . . . . . 12 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) = (Base‘𝐺))
8 isga.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
97, 8syl6eqr 2812 . . . . . . . . . . 11 ((𝑔 = 𝐺𝑠 = 𝑌) → (Base‘𝑔) = 𝑋)
105, 9sylan9eqr 2816 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑏 = 𝑋)
1110, 4xpeq12d 5297 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑏 × 𝑠) = (𝑋 × 𝑌))
124, 11oveq12d 6831 . . . . . . . 8 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑠𝑚 (𝑏 × 𝑠)) = (𝑌𝑚 (𝑋 × 𝑌)))
13 simpll 807 . . . . . . . . . . . . . 14 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑔 = 𝐺)
1413fveq2d 6356 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g𝑔) = (0g𝐺))
15 isga.3 . . . . . . . . . . . . 13 0 = (0g𝐺)
1614, 15syl6eqr 2812 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g𝑔) = 0 )
1716oveq1d 6828 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((0g𝑔)𝑚𝑥) = ( 0 𝑚𝑥))
1817eqeq1d 2762 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((0g𝑔)𝑚𝑥) = 𝑥 ↔ ( 0 𝑚𝑥) = 𝑥))
1913fveq2d 6356 . . . . . . . . . . . . . . . 16 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g𝑔) = (+g𝐺))
20 isga.2 . . . . . . . . . . . . . . . 16 + = (+g𝐺)
2119, 20syl6eqr 2812 . . . . . . . . . . . . . . 15 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g𝑔) = + )
2221oveqd 6830 . . . . . . . . . . . . . 14 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑦(+g𝑔)𝑧) = (𝑦 + 𝑧))
2322oveq1d 6828 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((𝑦(+g𝑔)𝑧)𝑚𝑥) = ((𝑦 + 𝑧)𝑚𝑥))
2423eqeq1d 2762 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2510, 24raleqbidv 3291 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2610, 25raleqbidv 3291 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
2718, 26anbi12d 749 . . . . . . . . 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 3291 . . . . . . . 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 3335 . . . . . . 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 3701 . . . . . 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 6841 . . . . . . 7 (𝑌𝑚 (𝑋 × 𝑌)) ∈ V
3231rabex 4964 . . . . . 6 {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} ∈ V
3330, 1, 32ovmpt2a 6956 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝐺 GrpAct 𝑌) = {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
3433eleq2d 2825 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ∈ {𝑚 ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∣ ∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))}))
35 oveq 6819 . . . . . . . 8 (𝑚 = → ( 0 𝑚𝑥) = ( 0 𝑥))
3635eqeq1d 2762 . . . . . . 7 (𝑚 = → (( 0 𝑚𝑥) = 𝑥 ↔ ( 0 𝑥) = 𝑥))
37 oveq 6819 . . . . . . . . 9 (𝑚 = → ((𝑦 + 𝑧)𝑚𝑥) = ((𝑦 + 𝑧) 𝑥))
38 oveq 6819 . . . . . . . . . 10 (𝑚 = → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 (𝑧𝑚𝑥)))
39 oveq 6819 . . . . . . . . . . 11 (𝑚 = → (𝑧𝑚𝑥) = (𝑧 𝑥))
4039oveq2d 6829 . . . . . . . . . 10 (𝑚 = → (𝑦 (𝑧𝑚𝑥)) = (𝑦 (𝑧 𝑥)))
4138, 40eqtrd 2794 . . . . . . . . 9 (𝑚 = → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 (𝑧 𝑥)))
4237, 41eqeq12d 2775 . . . . . . . 8 (𝑚 = → (((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
43422ralbidv 3127 . . . . . . 7 (𝑚 = → (∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
4436, 43anbi12d 749 . . . . . 6 (𝑚 = → ((( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
4544ralbidv 3124 . . . . 5 (𝑚 = → (∀𝑥𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
4645elrab 3504 . . . 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, 46syl6bb 276 . . 3 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ( ∈ (𝑌𝑚 (𝑋 × 𝑌)) ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
48 simpr 479 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
49 fvex 6362 . . . . . . 7 (Base‘𝐺) ∈ V
508, 49eqeltri 2835 . . . . . 6 𝑋 ∈ V
51 xpexg 7125 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
5250, 48, 51sylancr 698 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
5348, 52elmapd 8037 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝑌𝑚 (𝑋 × 𝑌)) ↔ :(𝑋 × 𝑌)⟶𝑌))
5453anbi1d 743 . . 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))))))
5547, 54bitrd 268 . 2 ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ∈ (𝐺 GrpAct 𝑌) ↔ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
562, 55biadan2 677 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 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  csb 3674   × cxp 5264  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  Basecbs 16059  +gcplusg 16143  0gc0g 16302  Grpcgrp 17623   GrpAct cga 17922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-ga 17923
This theorem is referenced by:  gagrp  17925  gaset  17926  gagrpid  17927  gaf  17928  gaass  17930  ga0  17931  gaid  17932  subgga  17933  gass  17934  gasubg  17935  lactghmga  18024  sylow1lem2  18214  sylow2blem2  18236  sylow3lem1  18242
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