Theorem isga 19223

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 19238). 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 19222	. . 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 7630	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
3		fvexd 6873	. . . . . . 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 6862	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) = (Base‘𝐺))
8		isga.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2782	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) → (Base‘𝑔) = 𝑋)
10	5, 9	sylan9eqr 2786	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → 𝑏 = 𝑋)
11	10, 4	xpeq12d 5669	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑏 × 𝑠) = (𝑋 × 𝑌))
12	4, 11	oveq12d 7405	. . . . . . . 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 6862	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g‘𝑔) = (0g‘𝐺))
15		isga.3	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝐺)
16	14, 15	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (0g‘𝑔) = 0 )
17	16	oveq1d 7402	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((0g‘𝑔)𝑚𝑥) = ( 0 𝑚𝑥))
18	17	eqeq1d 2731	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((0g‘𝑔)𝑚𝑥) = 𝑥 ↔ ( 0 𝑚𝑥) = 𝑥))
19	13	fveq2d 6862	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g‘𝑔) = (+g‘𝐺))
20		isga.2	. . . . . . . . . . . . . . . 16 ⊢ + = (+g‘𝐺)
21	19, 20	eqtr4di 2782	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (+g‘𝑔) = + )
22	21	oveqd 7404	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (𝑦(+g‘𝑔)𝑧) = (𝑦 + 𝑧))
23	22	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = ((𝑦 + 𝑧)𝑚𝑥))
24	23	eqeq1d 2731	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑠 = 𝑌) ∧ 𝑏 = (Base‘𝑔)) → (((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥))))
25	10, 24	raleqbidv 3319	. . . . . . . . . . 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 3319	. . . . . . . . . 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 3319	. . . . . . . 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 3424	. . . . . . 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 3898	. . . . . 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 7420	. . . . . . 7 ⊢ (𝑌 ↑m (𝑋 × 𝑌)) ∈ V
32	31	rabex 5294	. . . . . 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 7544	. . . . 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 2814	. . . 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 7393	. . . . . . . 8 ⊢ (𝑚 = ⊕ → ( 0 𝑚𝑥) = ( 0 ⊕ 𝑥))
36	35	eqeq1d 2731	. . . . . . 7 ⊢ (𝑚 = ⊕ → (( 0 𝑚𝑥) = 𝑥 ↔ ( 0 ⊕ 𝑥) = 𝑥))
37		oveq 7393	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → ((𝑦 + 𝑧)𝑚𝑥) = ((𝑦 + 𝑧) ⊕ 𝑥))
38		oveq 7393	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧𝑚𝑥)))
39		oveq 7393	. . . . . . . . . . 11 ⊢ (𝑚 = ⊕ → (𝑧𝑚𝑥) = (𝑧 ⊕ 𝑥))
40	39	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑚 = ⊕ → (𝑦 ⊕ (𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
41	38, 40	eqtrd 2764	. . . . . . . . 9 ⊢ (𝑚 = ⊕ → (𝑦𝑚(𝑧𝑚𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
42	37, 41	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑚 = ⊕ → (((𝑦 + 𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))
43	42	2ralbidv 3201	. . . . . . 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 3156	. . . . 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 3659	. . . 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 6872	. . . . . 6 ⊢ 𝑋 ∈ V
50		xpexg 7726	. . . . . 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 8813	. . . 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 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447  ⦋csb 3862   × cxp 5636  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Grpcgrp 18865   GrpAct cga 19221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-ga 19222

This theorem is referenced by:  gagrp  19224  gaset  19225  gagrpid  19226  gaf  19227  gaass  19229  ga0  19230  gaid  19231  subgga  19232  gass  19233  gasubg  19234  lactghmga  19335  sylow1lem2  19529  sylow2blem2  19551  sylow3lem1  19557  conjga  33127