Theorem isga 19204

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 19219). 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 19203	. . 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 7587	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
3		fvexd 6837	. . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → 𝑔 = 𝐺)
7	6	fveq2d 6826	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) = (Base‘𝐺))
8		isga.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2784	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) = 𝑋)
10	5, 9	sylan9eqr 2788	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑏 = 𝑋)
11	10, 4	xpeq12d 5647	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑏 × 𝑠) = (𝑋 × 𝑌))
12	4, 11	oveq12d 7364	. . . . . . . 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 6826	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g‘𝑔) = (0g‘𝐺))
15		isga.3	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝐺)
16	14, 15	eqtr4di 2784	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g‘𝑔) = 0 )
17	16	oveq1d 7361	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((0g‘𝑔)𝑚𝑥) = ( 0 𝑚𝑥))
18	17	eqeq1d 2733	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((0g‘𝑔)𝑚𝑥) = 𝑥 ↔ ( 0 𝑚𝑥) = 𝑥))
19	13	fveq2d 6826	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g‘𝑔) = (+g‘𝐺))
20		isga.2	. . . . . . . . . . . . . . . 16 ⊢ + = (+g‘𝐺)
21	19, 20	eqtr4di 2784	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g‘𝑔) = + )
22	21	oveqd 7363	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑦(+g‘𝑔)𝑧) = (𝑦 + 𝑧))
23	22	oveq1d 7361	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = ((𝑦 + 𝑧)𝑚𝑥))
24	23	eqeq1d 2733	. . . . . . . . . . . 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 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 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 3413	. . . . . . 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 3886	. . . . . 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 7379	. . . . . . 7 ⊢ (𝑌 ↑m (𝑋 × 𝑌)) ∈ V
32	31	rabex 5277	. . . . . 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 7501	. . . . 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 2817	. . . 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 7352	. . . . . . . 8 ⊢ (𝑚 = ⊕ → ( 0 𝑚𝑥) = ( 0 ⊕ 𝑥))
36	35	eqeq1d 2733	. . . . . . 7 ⊢ (𝑚 = ⊕ → (( 0 𝑚𝑥) = 𝑥 ↔ ( 0 ⊕ 𝑥) = 𝑥))
37		oveq 7352	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → ((𝑦 + 𝑧)𝑚𝑥) = ((𝑦 + 𝑧) ⊕ 𝑥))
38		oveq 7352	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧𝑚𝑥)))
39		oveq 7352	. . . . . . . . . . 11 ⊢ (𝑚 = ⊕ → (𝑧𝑚𝑥) = (𝑧 ⊕ 𝑥))
40	39	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦 ⊕ (𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
41	38, 40	eqtrd 2766	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
42	37, 41	eqeq12d 2747	. . . . . . . 8 ⊢ (𝑚 = ⊕ → (((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))
43	42	2ralbidv 3196	. . . . . . 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 3155	. . . . 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 3647	. . . 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 6836	. . . . . 6 ⊢ 𝑋 ∈ V
50		xpexg 7683	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
51	49, 48, 50	sylancr 587	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → (𝑋 × 𝑌) ∈ V)
52	48, 51	elmapd 8764	. . . 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436  ⦋csb 3850   × cxp 5614  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  Basecbs 17120  +_gcplusg 17161  0_gc0g 17343  Grpcgrp 18846   GrpAct cga 19202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-ga 19203

This theorem is referenced by:  gagrp  19205  gaset  19206  gagrpid  19207  gaf  19208  gaass  19210  ga0  19211  gaid  19212  subgga  19213  gass  19214  gasubg  19215  lactghmga  19318  sylow1lem2  19512  sylow2blem2  19534  sylow3lem1  19540  conjga  33137  mplvrpmga  33573