Theorem isga 19155

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 19170). 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 19154	. . 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 7648	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
3		fvexd 6907	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) ∈ V)
4		simplr 768	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑠 = 𝑌)
5		id 22	. . . . . . . . . . 11 ⊢ (𝑏 = (Base‘𝑔) → 𝑏 = (Base‘𝑔))
6		simpl 484	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → 𝑔 = 𝐺)
7	6	fveq2d 6896	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) = (Base‘𝐺))
8		isga.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2791	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) = 𝑋)
10	5, 9	sylan9eqr 2795	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑏 = 𝑋)
11	10, 4	xpeq12d 5708	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑏 × 𝑠) = (𝑋 × 𝑌))
12	4, 11	oveq12d 7427	. . . . . . . 8 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑠 ↑m (𝑏 × 𝑠)) = (𝑌 ↑m (𝑋 × 𝑌)))
13		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑔 = 𝐺)
14	13	fveq2d 6896	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g‘𝑔) = (0g‘𝐺))
15		isga.3	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝐺)
16	14, 15	eqtr4di 2791	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g‘𝑔) = 0 )
17	16	oveq1d 7424	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((0g‘𝑔)𝑚𝑥) = ( 0 𝑚𝑥))
18	17	eqeq1d 2735	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((0g‘𝑔)𝑚𝑥) = 𝑥 ↔ ( 0 𝑚𝑥) = 𝑥))
19	13	fveq2d 6896	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g‘𝑔) = (+g‘𝐺))
20		isga.2	. . . . . . . . . . . . . . . 16 ⊢ + = (+g‘𝐺)
21	19, 20	eqtr4di 2791	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g‘𝑔) = + )
22	21	oveqd 7426	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑦(+g‘𝑔)𝑧) = (𝑦 + 𝑧))
23	22	oveq1d 7424	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = ((𝑦 + 𝑧)𝑚𝑥))
24	23	eqeq1d 2735	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
25	10, 24	raleqbidv 3343	. . . . . . . . . . 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 3343	. . . . . . . . . 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 632	. . . . . . . . 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 3343	. . . . . . . 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 3450	. . . . . . 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 3932	. . . . . 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 7442	. . . . . . 7 ⊢ (𝑌 ↑m (𝑋 × 𝑌)) ∈ V
32	31	rabex 5333	. . . . . 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 7563	. . . . 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 2820	. . . 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 7415	. . . . . . . 8 ⊢ (𝑚 = ⊕ → ( 0 𝑚𝑥) = ( 0 ⊕ 𝑥))
36	35	eqeq1d 2735	. . . . . . 7 ⊢ (𝑚 = ⊕ → (( 0 𝑚𝑥) = 𝑥 ↔ ( 0 ⊕ 𝑥) = 𝑥))
37		oveq 7415	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → ((𝑦 + 𝑧)𝑚𝑥) = ((𝑦 + 𝑧) ⊕ 𝑥))
38		oveq 7415	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧𝑚𝑥)))
39		oveq 7415	. . . . . . . . . . 11 ⊢ (𝑚 = ⊕ → (𝑧𝑚𝑥) = (𝑧 ⊕ 𝑥))
40	39	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦 ⊕ (𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
41	38, 40	eqtrd 2773	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
42	37, 41	eqeq12d 2749	. . . . . . . 8 ⊢ (𝑚 = ⊕ → (((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))
43	42	2ralbidv 3219	. . . . . . 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 632	. . . . . 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 3178	. . . . 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 3684	. . . 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 486	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
49	8	fvexi 6906	. . . . . 6 ⊢ 𝑋 ∈ V
50		xpexg 7737	. . . . . 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 8834	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ⊕ ∈ (𝑌 ↑m (𝑋 × 𝑌)) ↔ ⊕ :(𝑋 × 𝑌)⟶𝑌))
53	52	anbi1d 631	. . 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 821	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 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475  ⦋csb 3894   × cxp 5675  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409   ↑_m cmap 8820  Basecbs 17144  +_gcplusg 17197  0_gc0g 17385  Grpcgrp 18819   GrpAct cga 19153

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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-ga 19154

This theorem is referenced by:  gagrp  19156  gaset  19157  gagrpid  19158  gaf  19159  gaass  19161  ga0  19162  gaid  19163  subgga  19164  gass  19165  gasubg  19166  lactghmga  19273  sylow1lem2  19467  sylow2blem2  19489  sylow3lem1  19495