Theorem isga 19149

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 19164). 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 19148	. . 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 7644	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
3		fvexd 6903	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) ∈ V)
4		simplr 767	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑠 = 𝑌)
5		id 22	. . . . . . . . . . 11 ⊢ (𝑏 = (Base‘𝑔) → 𝑏 = (Base‘𝑔))
6		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → 𝑔 = 𝐺)
7	6	fveq2d 6892	. . . . . . . . . . . 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 5706	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑏 × 𝑠) = (𝑋 × 𝑌))
12	4, 11	oveq12d 7423	. . . . . . . 8 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑠 ↑m (𝑏 × 𝑠)) = (𝑌 ↑m (𝑋 × 𝑌)))
13		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑔 = 𝐺)
14	13	fveq2d 6892	. . . . . . . . . . . . 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 7420	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((0g‘𝑔)𝑚𝑥) = ( 0 𝑚𝑥))
18	17	eqeq1d 2734	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((0g‘𝑔)𝑚𝑥) = 𝑥 ↔ ( 0 𝑚𝑥) = 𝑥))
19	13	fveq2d 6892	. . . . . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑦(+g‘𝑔)𝑧) = (𝑦 + 𝑧))
23	22	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = ((𝑦 + 𝑧)𝑚𝑥))
24	23	eqeq1d 2734	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
25	10, 24	raleqbidv 3342	. . . . . . . . . . 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 3342	. . . . . . . . . 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 631	. . . . . . . . 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 3342	. . . . . . . 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 3449	. . . . . . 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 3930	. . . . . 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 7438	. . . . . . 7 ⊢ (𝑌 ↑m (𝑋 × 𝑌)) ∈ V
32	31	rabex 5331	. . . . . 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 7559	. . . . 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 2819	. . . 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 7411	. . . . . . . 8 ⊢ (𝑚 = ⊕ → ( 0 𝑚𝑥) = ( 0 ⊕ 𝑥))
36	35	eqeq1d 2734	. . . . . . 7 ⊢ (𝑚 = ⊕ → (( 0 𝑚𝑥) = 𝑥 ↔ ( 0 ⊕ 𝑥) = 𝑥))
37		oveq 7411	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → ((𝑦 + 𝑧)𝑚𝑥) = ((𝑦 + 𝑧) ⊕ 𝑥))
38		oveq 7411	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧𝑚𝑥)))
39		oveq 7411	. . . . . . . . . . 11 ⊢ (𝑚 = ⊕ → (𝑧𝑚𝑥) = (𝑧 ⊕ 𝑥))
40	39	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦 ⊕ (𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
41	38, 40	eqtrd 2772	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
42	37, 41	eqeq12d 2748	. . . . . . . 8 ⊢ (𝑚 = ⊕ → (((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))
43	42	2ralbidv 3218	. . . . . . 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 631	. . . . . 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 3177	. . . . 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 3682	. . . 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 286	. . 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 485	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
49	8	fvexi 6902	. . . . . 6 ⊢ 𝑋 ∈ V
50		xpexg 7733	. . . . . 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 8830	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) → ( ⊕ ∈ (𝑌 ↑m (𝑋 × 𝑌)) ↔ ⊕ :(𝑋 × 𝑌)⟶𝑌))
53	52	anbi1d 630	. . 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 278	. 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 820	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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  {crab 3432  Vcvv 3474  ⦋csb 3892   × cxp 5673  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   ↑_m cmap 8816  Basecbs 17140  +_gcplusg 17193  0_gc0g 17381  Grpcgrp 18815   GrpAct cga 19147

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-ga 19148

This theorem is referenced by:  gagrp  19150  gaset  19151  gagrpid  19152  gaf  19153  gaass  19155  ga0  19156  gaid  19157  subgga  19158  gass  19159  gasubg  19160  lactghmga  19267  sylow1lem2  19461  sylow2blem2  19483  sylow3lem1  19489