Theorem isga 19257

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 19272). 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	⊢ + = (+g‘𝐺)
isga.3	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
isga	⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((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.

Step	Hyp	Ref	Expression
1		df-ga 19256	. . 3 ⊢ GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ ⦋(Base‘𝑔) / 𝑏⦌{𝑚 ∈ (𝑠 ↑m (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
2	1	elmpocl 7601	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
3		fvexd 6849	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) ∈ V)
4		simplr 769	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑠 = 𝑌)
5		id 22	. . . . . . . . . . 11 ⊢ (𝑏 = (Base‘𝑔) → 𝑏 = (Base‘𝑔))
6		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → 𝑔 = 𝐺)
7	6	fveq2d 6838	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) = (Base‘𝐺))
8		isga.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2790	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) = 𝑋)
10	5, 9	sylan9eqr 2794	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑏 = 𝑋)
11	10, 4	xpeq12d 5655	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑏 × 𝑠) = (𝑋 × 𝑌))
12	4, 11	oveq12d 7378	. . . . . . . 8 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑠 ↑m (𝑏 × 𝑠)) = (𝑌 ↑m (𝑋 × 𝑌)))
13		simpll 767	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑔 = 𝐺)
14	13	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g‘𝑔) = (0g‘𝐺))
15		isga.3	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝐺)
16	14, 15	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g‘𝑔) = 0 )
17	16	oveq1d 7375	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((0g‘𝑔)𝑚𝑥) = ( 0 𝑚𝑥))
18	17	eqeq1d 2739	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((0g‘𝑔)𝑚𝑥) = 𝑥 ↔ ( 0 𝑚𝑥) = 𝑥))
19	13	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g‘𝑔) = (+g‘𝐺))
20		isga.2	. . . . . . . . . . . . . . . 16 ⊢ + = (+g‘𝐺)
21	19, 20	eqtr4di 2790	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g‘𝑔) = + )
22	21	oveqd 7377	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑦(+g‘𝑔)𝑧) = (𝑦 + 𝑧))
23	22	oveq1d 7375	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = ((𝑦 + 𝑧)𝑚𝑥))
24	23	eqeq1d 2739	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
25	10, 24	raleqbidv 3312	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
26	10, 25	raleqbidv 3312	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
27	18, 26	anbi12d 633	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))))
28	4, 27	raleqbidv 3312	. . . . . . . 8 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ ∀𝑥 ∈ 𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))))
29	12, 28	rabeqbidv 3408	. . . . . . 7 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → {𝑚 ∈ (𝑠 ↑m (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} = {𝑚 ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∣ ∀𝑥 ∈ 𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
30	3, 29	csbied 3874	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → ⦋(Base‘𝑔) / 𝑏⦌{𝑚 ∈ (𝑠 ↑m (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} = {𝑚 ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∣ ∀𝑥 ∈ 𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
31		ovex 7393	. . . . . . 7 ⊢ (𝑌 ↑m (𝑋 × 𝑌)) ∈ V
32	31	rabex 5276	. . . . . 6 ⊢ {𝑚 ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∣ ∀𝑥 ∈ 𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} ∈ V
33	30, 1, 32	ovmpoa 7515	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝐺 GrpAct 𝑌) = {𝑚 ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∣ ∀𝑥 ∈ 𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
34	33	eleq2d 2823	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ⊕ ∈ {𝑚 ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∣ ∀𝑥 ∈ 𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))}))
35		oveq 7366	. . . . . . . 8 ⊢ (𝑚 = ⊕ → ( 0 𝑚𝑥) = ( 0 ⊕ 𝑥))
36	35	eqeq1d 2739	. . . . . . 7 ⊢ (𝑚 = ⊕ → (( 0 𝑚𝑥) = 𝑥 ↔ ( 0 ⊕ 𝑥) = 𝑥))
37		oveq 7366	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → ((𝑦 + 𝑧)𝑚𝑥) = ((𝑦 + 𝑧) ⊕ 𝑥))
38		oveq 7366	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧𝑚𝑥)))
39		oveq 7366	. . . . . . . . . . 11 ⊢ (𝑚 = ⊕ → (𝑧𝑚𝑥) = (𝑧 ⊕ 𝑥))
40	39	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦 ⊕ (𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
41	38, 40	eqtrd 2772	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
42	37, 41	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑚 = ⊕ → (((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))
43	42	2ralbidv 3202	. . . . . . 7 ⊢ (𝑚 = ⊕ → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))
44	36, 43	anbi12d 633	. . . . . 6 ⊢ (𝑚 = ⊕ → ((( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))
45	44	ralbidv 3161	. . . . 5 ⊢ (𝑚 = ⊕ → (∀𝑥 ∈ 𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))) ↔ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))
46	45	elrab 3635	. . . 4 ⊢ ( ⊕ ∈ {𝑚 ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∣ ∀𝑥 ∈ 𝑌 (( 0 𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))} ↔ ( ⊕ ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))
47	34, 46	bitrdi 287	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ( ⊕ ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
48		simpr 484	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
49	8	fvexi 6848	. . . . . 6 ⊢ 𝑋 ∈ V
50		xpexg 7697	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
51	49, 48, 50	sylancr 588	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
52	48, 51	elmapd 8780	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ⊕ ∈ (𝑌 ↑m (𝑋 × 𝑌)) ↔ ⊕ :(𝑋 × 𝑌)⟶𝑌))
53	52	anbi1d 632	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (( ⊕ ∈ (𝑌 ↑m (𝑋 × 𝑌)) ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))) ↔ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
54	47, 53	bitrd 279	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
55	2, 54	biadanii 822	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  Vcvv 3430  ⦋csb 3838   × cxp 5622  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766  Basecbs 17170  +_gcplusg 17211  0_gc0g 17393  Grpcgrp 18900   GrpAct cga 19255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-ga 19256

This theorem is referenced by:  gagrp  19258  gaset  19259  gagrpid  19260  gaf  19261  gaass  19263  ga0  19264  gaid  19265  subgga  19266  gass  19267  gasubg  19268  lactghmga  19371  sylow1lem2  19565  sylow2blem2  19587  sylow3lem1  19593  conjga  33246  mplvrpmga  33704