Theorem sylow1lem2 18220

Description: Lemma for sylow1 18224. The function ⊕ is a group action on 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
sylow1.x	⊢ 𝑋 = (Base‘𝐺)
sylow1.g	⊢ (𝜑 → 𝐺 ∈ Grp)
sylow1.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow1.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
sylow1.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
sylow1.d	⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))
sylow1lem.a	⊢ + = (+g‘𝐺)
sylow1lem.s	⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}
sylow1lem.m	⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))

Assertion

Ref	Expression
sylow1lem2	⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))

Distinct variable groups:   𝑥,𝑠,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑁,𝑠,𝑥,𝑦,𝑧   𝑋,𝑠,𝑥,𝑦,𝑧   + ,𝑠,𝑥,𝑦,𝑧   𝑥, ⊕ ,𝑦,𝑧   𝐺,𝑠,𝑥,𝑦,𝑧   𝑃,𝑠,𝑥,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑠)   ⊕ (𝑠)   𝑆(𝑠)

Proof of Theorem sylow1lem2

Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		sylow1lem.s	. . . 4 ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}
3		sylow1.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
4		fvex 6342	. . . . . 6 ⊢ (Base‘𝐺) ∈ V
5	3, 4	eqeltri 2846	. . . . 5 ⊢ 𝑋 ∈ V
6	5	pwex 4979	. . . 4 ⊢ 𝒫 𝑋 ∈ V
7	2, 6	rabex2 4948	. . 3 ⊢ 𝑆 ∈ V
8	1, 7	jctir 504	. 2 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ 𝑆 ∈ V))
9		simprl 746	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑋)
10		sylow1lem.a	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝐺)
11		eqid 2771	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧))
12	3, 10, 11	grplmulf1o 17696	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1-onto→𝑋)
13	1, 9, 12	syl2an2r 656	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1-onto→𝑋)
14		f1of1 6277	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1-onto→𝑋 → (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1→𝑋)
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1→𝑋)
16		simprr 748	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
17		fveq2 6332	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑦 → (♯‘𝑠) = (♯‘𝑦))
18	17	eqeq1d 2773	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑦 → ((♯‘𝑠) = (𝑃↑𝑁) ↔ (♯‘𝑦) = (𝑃↑𝑁)))
19	18, 2	elrab2 3518	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (♯‘𝑦) = (𝑃↑𝑁)))
20	16, 19	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑦 ∈ 𝒫 𝑋 ∧ (♯‘𝑦) = (𝑃↑𝑁)))
21	20	simpld 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝒫 𝑋)
22	21	elpwid 4309	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ⊆ 𝑋)
23		f1ssres 6248	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1→𝑋 ∧ 𝑦 ⊆ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦):𝑦–1-1→𝑋)
24	15, 22, 23	syl2anc 565	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦):𝑦–1-1→𝑋)
25		resmpt 5590	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
26		f1eq1 6236	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) → (((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦):𝑦–1-1→𝑋 ↔ (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋))
27	22, 25, 26	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦):𝑦–1-1→𝑋 ↔ (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋))
28	24, 27	mpbid 222	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋)
29		f1f 6241	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋 → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦⟶𝑋)
30		frn 6193	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦⟶𝑋 → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋)
31	28, 29, 30	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋)
32	5	elpw2 4959	. . . . . . 7 ⊢ (ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝒫 𝑋 ↔ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋)
33	31, 32	sylibr 224	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝒫 𝑋)
34		f1f1orn 6289	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋 → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1-onto→ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
35		vex 3354	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
36	35	f1oen 8129	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1-onto→ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) → 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
37	28, 34, 36	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
38		sylow1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ Fin)
39		ssfi 8335	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Fin ∧ 𝑦 ⊆ 𝑋) → 𝑦 ∈ Fin)
40	38, 22, 39	syl2an2r 656	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ Fin)
41		ssfi 8335	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Fin ∧ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ Fin)
42	38, 31, 41	syl2an2r 656	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ Fin)
43		hashen 13338	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ Fin) → ((♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ↔ 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
44	40, 42, 43	syl2anc 565	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ↔ 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
45	37, 44	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
46	20	simprd 477	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘𝑦) = (𝑃↑𝑁))
47	45, 46	eqtr3d 2807	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) = (𝑃↑𝑁))
48		fveq2 6332	. . . . . . . 8 ⊢ (𝑠 = ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) → (♯‘𝑠) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
49	48	eqeq1d 2773	. . . . . . 7 ⊢ (𝑠 = ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) → ((♯‘𝑠) = (𝑃↑𝑁) ↔ (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) = (𝑃↑𝑁)))
50	49, 2	elrab2 3518	. . . . . 6 ⊢ (ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆 ↔ (ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝒫 𝑋 ∧ (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) = (𝑃↑𝑁)))
51	33, 47, 50	sylanbrc 564	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆)
52	51	ralrimivva 3120	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆)
53		sylow1lem.m	. . . . 5 ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
54	53	fmpt2 7386	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆 ↔ ⊕ :(𝑋 × 𝑆)⟶𝑆)
55	52, 54	sylib 208	. . 3 ⊢ (𝜑 → ⊕ :(𝑋 × 𝑆)⟶𝑆)
56	1	adantr 466	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐺 ∈ Grp)
57		eqid 2771	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
58	3, 57	grpidcl 17657	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
59	56, 58	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0g‘𝐺) ∈ 𝑋)
60		simpr 471	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
61		simpr 471	. . . . . . . . . 10 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
62		simpl 468	. . . . . . . . . . 11 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → 𝑥 = (0g‘𝐺))
63	62	oveq1d 6807	. . . . . . . . . 10 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((0g‘𝐺) + 𝑧))
64	61, 63	mpteq12dv 4867	. . . . . . . . 9 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)))
65	64	rneqd 5491	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)))
66		vex 3354	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
67	66	mptex 6629	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) ∈ V
68	67	rnex 7246	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) ∈ V
69	65, 53, 68	ovmpt2a 6937	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → ((0g‘𝐺) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)))
70	59, 60, 69	syl2anc 565	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0g‘𝐺) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)))
71		ssrab2 3836	. . . . . . . . . . . . . . 15 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ⊆ 𝒫 𝑋
72	2, 71	eqsstri 3784	. . . . . . . . . . . . . 14 ⊢ 𝑆 ⊆ 𝒫 𝑋
73	72, 60	sseldi 3750	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝒫 𝑋)
74	73	elpwid 4309	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ⊆ 𝑋)
75	74	sselda 3752	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑧 ∈ 𝑎) → 𝑧 ∈ 𝑋)
76	3, 10, 57	grplid 17659	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺) + 𝑧) = 𝑧)
77	56, 75, 76	syl2an2r 656	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑧 ∈ 𝑎) → ((0g‘𝐺) + 𝑧) = 𝑧)
78	77	mpteq2dva 4878	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) = (𝑧 ∈ 𝑎 ↦ 𝑧))
79		mptresid 5597	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑎 ↦ 𝑧) = ( I ↾ 𝑎)
80	78, 79	syl6eq 2821	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) = ( I ↾ 𝑎))
81	80	rneqd 5491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) = ran ( I ↾ 𝑎))
82		rnresi 5620	. . . . . . 7 ⊢ ran ( I ↾ 𝑎) = 𝑎
83	81, 82	syl6eq 2821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) = 𝑎)
84	70, 83	eqtrd 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0g‘𝐺) ⊕ 𝑎) = 𝑎)
85		ovex 6822	. . . . . . . . . 10 ⊢ (𝑐 + 𝑧) ∈ V
86		oveq2 6800	. . . . . . . . . 10 ⊢ (𝑤 = (𝑐 + 𝑧) → (𝑏 + 𝑤) = (𝑏 + (𝑐 + 𝑧)))
87	85, 86	abrexco 6644	. . . . . . . . 9 ⊢ {𝑢 ∣ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)}𝑢 = (𝑏 + 𝑤)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = (𝑏 + (𝑐 + 𝑧))}
88		simprr 748	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑐 ∈ 𝑋)
89	60	adantr 466	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑎 ∈ 𝑆)
90		simpr 471	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
91		simpl 468	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → 𝑥 = 𝑐)
92	91	oveq1d 6807	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = (𝑐 + 𝑧))
93	90, 92	mpteq12dv 4867	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
94	93	rneqd 5491	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
95	66	mptex 6629	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) ∈ V
96	95	rnex 7246	. . . . . . . . . . . . . 14 ⊢ ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) ∈ V
97	94, 53, 96	ovmpt2a 6937	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → (𝑐 ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
98	88, 89, 97	syl2anc 565	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
99		eqid 2771	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧))
100	99	rnmpt 5509	. . . . . . . . . . . 12 ⊢ ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) = {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)}
101	98, 100	syl6eq 2821	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) = {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)})
102	101	rexeqdv 3294	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤) ↔ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)}𝑢 = (𝑏 + 𝑤)))
103	102	abbidv 2890	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤)} = {𝑢 ∣ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)}𝑢 = (𝑏 + 𝑤)})
104	56	ad2antrr 697	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝐺 ∈ Grp)
105		simprl 746	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
106	105	adantr 466	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑏 ∈ 𝑋)
107	88	adantr 466	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑐 ∈ 𝑋)
108	75	adantlr 686	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑧 ∈ 𝑋)
109	3, 10	grpass 17638	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧)))
110	104, 106, 107, 108, 109	syl13anc 1478	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧)))
111	110	eqeq2d 2781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (𝑢 = ((𝑏 + 𝑐) + 𝑧) ↔ 𝑢 = (𝑏 + (𝑐 + 𝑧))))
112	111	rexbidva 3197	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧) ↔ ∃𝑧 ∈ 𝑎 𝑢 = (𝑏 + (𝑐 + 𝑧))))
113	112	abbidv 2890	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = (𝑏 + (𝑐 + 𝑧))})
114	87, 103, 113	3eqtr4a 2831	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧)})
115		eqid 2771	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤))
116	115	rnmpt 5509	. . . . . . . 8 ⊢ ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) = {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤)}
117		eqid 2771	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧))
118	117	rnmpt 5509	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧)}
119	114, 116, 118	3eqtr4g 2830	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
120	55	ad2antrr 697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⊕ :(𝑋 × 𝑆)⟶𝑆)
121	120, 88, 89	fovrnd 6952	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) ∈ 𝑆)
122		simpr 471	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → 𝑦 = (𝑐 ⊕ 𝑎))
123		simpl 468	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → 𝑥 = 𝑏)
124	123	oveq1d 6807	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑥 + 𝑧) = (𝑏 + 𝑧))
125	122, 124	mpteq12dv 4867	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑧)))
126		oveq2 6800	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑏 + 𝑧) = (𝑏 + 𝑤))
127	126	cbvmptv 4884	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑧)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤))
128	125, 127	syl6eq 2821	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
129	128	rneqd 5491	. . . . . . . . 9 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
130		ovex 6822	. . . . . . . . . . 11 ⊢ (𝑐 ⊕ 𝑎) ∈ V
131	130	mptex 6629	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) ∈ V
132	131	rnex 7246	. . . . . . . . 9 ⊢ ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) ∈ V
133	129, 53, 132	ovmpt2a 6937	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝑋 ∧ (𝑐 ⊕ 𝑎) ∈ 𝑆) → (𝑏 ⊕ (𝑐 ⊕ 𝑎)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
134	105, 121, 133	syl2anc 565	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑏 ⊕ (𝑐 ⊕ 𝑎)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
135	1	ad2antrr 697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐺 ∈ Grp)
136	3, 10	grpcl 17637	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋) → (𝑏 + 𝑐) ∈ 𝑋)
137	135, 105, 88, 136	syl3anc 1476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑏 + 𝑐) ∈ 𝑋)
138		simpr 471	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
139		simpl 468	. . . . . . . . . . . 12 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑥 = (𝑏 + 𝑐))
140	139	oveq1d 6807	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((𝑏 + 𝑐) + 𝑧))
141	138, 140	mpteq12dv 4867	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
142	141	rneqd 5491	. . . . . . . . 9 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
143	66	mptex 6629	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) ∈ V
144	143	rnex 7246	. . . . . . . . 9 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) ∈ V
145	142, 53, 144	ovmpt2a 6937	. . . . . . . 8 ⊢ (((𝑏 + 𝑐) ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → ((𝑏 + 𝑐) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
146	137, 89, 145	syl2anc 565	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + 𝑐) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
147	119, 134, 146	3eqtr4rd 2816	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))
148	147	ralrimivva 3120	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))
149	84, 148	jca 495	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))
150	149	ralrimiva 3115	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))
151	55, 150	jca 495	. 2 ⊢ (𝜑 → ( ⊕ :(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑎 ∈ 𝑆 (((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))))
152	3, 10, 57	isga 17930	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) ↔ ((𝐺 ∈ Grp ∧ 𝑆 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑎 ∈ 𝑆 (((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))))
153	8, 151, 152	sylanbrc 564	1 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {cab 2757  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351   ⊆ wss 3723  𝒫 cpw 4297   class class class wbr 4786   ↦ cmpt 4863   I cid 5156   × cxp 5247  ran crn 5250   ↾ cres 5251  ⟶wf 6027  –1-1→wf1 6028  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794   ≈ cen 8105  Fincfn 8108  ℕ₀cn0 11493  ↑cexp 13066  ♯chash 13320   ∥ cdvds 15188  ℙcprime 15591  Basecbs 16063  +_gcplusg 16148  0_gc0g 16307  Grpcgrp 17629   GrpAct cga 17928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-n0 11494  df-z 11579  df-uz 11888  df-hash 13321  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-ga 17929

This theorem is referenced by:  sylow1lem3  18221  sylow1lem5  18223