Theorem sylow1lem2 18724

Description: Lemma for sylow1 18728. 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	3	fvexi 6684	. . . . 5 ⊢ 𝑋 ∈ V
5	4	pwex 5281	. . . 4 ⊢ 𝒫 𝑋 ∈ V
6	2, 5	rabex2 5237	. . 3 ⊢ 𝑆 ∈ V
7	1, 6	jctir 523	. 2 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ 𝑆 ∈ V))
8		simprl 769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑋)
9		sylow1lem.a	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝐺)
10		eqid 2821	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧))
11	3, 9, 10	grplmulf1o 18173	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1-onto→𝑋)
12	1, 8, 11	syl2an2r 683	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1-onto→𝑋)
13		f1of1 6614	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1-onto→𝑋 → (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1→𝑋)
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1→𝑋)
15		simprr 771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
16		fveqeq2 6679	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑦 → ((♯‘𝑠) = (𝑃↑𝑁) ↔ (♯‘𝑦) = (𝑃↑𝑁)))
17	16, 2	elrab2 3683	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (♯‘𝑦) = (𝑃↑𝑁)))
18	15, 17	sylib 220	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑦 ∈ 𝒫 𝑋 ∧ (♯‘𝑦) = (𝑃↑𝑁)))
19	18	simpld 497	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝒫 𝑋)
20	19	elpwid 4550	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ⊆ 𝑋)
21		f1ssres 6582	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)):𝑋–1-1→𝑋 ∧ 𝑦 ⊆ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦):𝑦–1-1→𝑋)
22	14, 20, 21	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦):𝑦–1-1→𝑋)
23		resmpt 5905	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
24		f1eq1 6570	. . . . . . . . . 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))
25	20, 23, 24	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))
26	22, 25	mpbid 234	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋)
27		f1f 6575	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋 → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦⟶𝑋)
28		frn 6520	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦⟶𝑋 → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋)
29	26, 27, 28	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋)
30	4	elpw2 5248	. . . . . . 7 ⊢ (ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝒫 𝑋 ↔ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋)
31	29, 30	sylibr 236	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝒫 𝑋)
32		f1f1orn 6626	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋 → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1-onto→ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
33		vex 3497	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
34	33	f1oen 8530	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1-onto→ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) → 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
35	26, 32, 34	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
36		sylow1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ Fin)
37		ssfi 8738	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Fin ∧ 𝑦 ⊆ 𝑋) → 𝑦 ∈ Fin)
38	36, 20, 37	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ Fin)
39		ssfi 8738	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Fin ∧ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ Fin)
40	36, 29, 39	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ Fin)
41		hashen 13708	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ Fin) → ((♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ↔ 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
42	38, 40, 41	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ↔ 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
43	35, 42	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
44	18	simprd 498	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘𝑦) = (𝑃↑𝑁))
45	43, 44	eqtr3d 2858	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) = (𝑃↑𝑁))
46		fveqeq2 6679	. . . . . . 7 ⊢ (𝑠 = ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) → ((♯‘𝑠) = (𝑃↑𝑁) ↔ (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) = (𝑃↑𝑁)))
47	46, 2	elrab2 3683	. . . . . 6 ⊢ (ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆 ↔ (ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝒫 𝑋 ∧ (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) = (𝑃↑𝑁)))
48	31, 45, 47	sylanbrc 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆)
49	48	ralrimivva 3191	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆)
50		sylow1lem.m	. . . . 5 ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
51	50	fmpo 7766	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆 ↔ ⊕ :(𝑋 × 𝑆)⟶𝑆)
52	49, 51	sylib 220	. . 3 ⊢ (𝜑 → ⊕ :(𝑋 × 𝑆)⟶𝑆)
53	1	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐺 ∈ Grp)
54		eqid 2821	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
55	3, 54	grpidcl 18131	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
56	53, 55	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0g‘𝐺) ∈ 𝑋)
57		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
58		simpr 487	. . . . . . . . . 10 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
59		simpl 485	. . . . . . . . . . 11 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → 𝑥 = (0g‘𝐺))
60	59	oveq1d 7171	. . . . . . . . . 10 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((0g‘𝐺) + 𝑧))
61	58, 60	mpteq12dv 5151	. . . . . . . . 9 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)))
62	61	rneqd 5808	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)))
63		vex 3497	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
64	63	mptex 6986	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) ∈ V
65	64	rnex 7617	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) ∈ V
66	62, 50, 65	ovmpoa 7305	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → ((0g‘𝐺) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)))
67	56, 57, 66	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0g‘𝐺) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)))
68	2	ssrab3 4057	. . . . . . . . . . . . . 14 ⊢ 𝑆 ⊆ 𝒫 𝑋
69	68, 57	sseldi 3965	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝒫 𝑋)
70	69	elpwid 4550	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ⊆ 𝑋)
71	70	sselda 3967	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑧 ∈ 𝑎) → 𝑧 ∈ 𝑋)
72	3, 9, 54	grplid 18133	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺) + 𝑧) = 𝑧)
73	53, 71, 72	syl2an2r 683	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑧 ∈ 𝑎) → ((0g‘𝐺) + 𝑧) = 𝑧)
74	73	mpteq2dva 5161	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) = (𝑧 ∈ 𝑎 ↦ 𝑧))
75		mptresid 5918	. . . . . . . . 9 ⊢ ( I ↾ 𝑎) = (𝑧 ∈ 𝑎 ↦ 𝑧)
76	74, 75	syl6eqr 2874	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) = ( I ↾ 𝑎))
77	76	rneqd 5808	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) = ran ( I ↾ 𝑎))
78		rnresi 5943	. . . . . . 7 ⊢ ran ( I ↾ 𝑎) = 𝑎
79	77, 78	syl6eq 2872	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ran (𝑧 ∈ 𝑎 ↦ ((0g‘𝐺) + 𝑧)) = 𝑎)
80	67, 79	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0g‘𝐺) ⊕ 𝑎) = 𝑎)
81		ovex 7189	. . . . . . . . . 10 ⊢ (𝑐 + 𝑧) ∈ V
82		oveq2 7164	. . . . . . . . . 10 ⊢ (𝑤 = (𝑐 + 𝑧) → (𝑏 + 𝑤) = (𝑏 + (𝑐 + 𝑧)))
83	81, 82	abrexco 7003	. . . . . . . . 9 ⊢ {𝑢 ∣ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)}𝑢 = (𝑏 + 𝑤)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = (𝑏 + (𝑐 + 𝑧))}
84		simprr 771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑐 ∈ 𝑋)
85	57	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑎 ∈ 𝑆)
86		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
87		simpl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → 𝑥 = 𝑐)
88	87	oveq1d 7171	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = (𝑐 + 𝑧))
89	86, 88	mpteq12dv 5151	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
90	89	rneqd 5808	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
91	63	mptex 6986	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) ∈ V
92	91	rnex 7617	. . . . . . . . . . . . . 14 ⊢ ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) ∈ V
93	90, 50, 92	ovmpoa 7305	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → (𝑐 ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
94	84, 85, 93	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
95		eqid 2821	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧))
96	95	rnmpt 5827	. . . . . . . . . . . 12 ⊢ ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) = {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)}
97	94, 96	syl6eq 2872	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) = {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)})
98	97	rexeqdv 3416	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤) ↔ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)}𝑢 = (𝑏 + 𝑤)))
99	98	abbidv 2885	. . . . . . . . 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)})
100	53	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝐺 ∈ Grp)
101		simprl 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
102	101	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑏 ∈ 𝑋)
103	84	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑐 ∈ 𝑋)
104	71	adantlr 713	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑧 ∈ 𝑋)
105	3, 9	grpass 18112	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧)))
106	100, 102, 103, 104, 105	syl13anc 1368	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧)))
107	106	eqeq2d 2832	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (𝑢 = ((𝑏 + 𝑐) + 𝑧) ↔ 𝑢 = (𝑏 + (𝑐 + 𝑧))))
108	107	rexbidva 3296	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧) ↔ ∃𝑧 ∈ 𝑎 𝑢 = (𝑏 + (𝑐 + 𝑧))))
109	108	abbidv 2885	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = (𝑏 + (𝑐 + 𝑧))})
110	83, 99, 109	3eqtr4a 2882	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧)})
111		eqid 2821	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤))
112	111	rnmpt 5827	. . . . . . . 8 ⊢ ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) = {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤)}
113		eqid 2821	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧))
114	113	rnmpt 5827	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧)}
115	110, 112, 114	3eqtr4g 2881	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
116	52	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⊕ :(𝑋 × 𝑆)⟶𝑆)
117	116, 84, 85	fovrnd 7320	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) ∈ 𝑆)
118		simpr 487	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → 𝑦 = (𝑐 ⊕ 𝑎))
119		simpl 485	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → 𝑥 = 𝑏)
120	119	oveq1d 7171	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑥 + 𝑧) = (𝑏 + 𝑧))
121	118, 120	mpteq12dv 5151	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑧)))
122		oveq2 7164	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑏 + 𝑧) = (𝑏 + 𝑤))
123	122	cbvmptv 5169	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑧)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤))
124	121, 123	syl6eq 2872	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
125	124	rneqd 5808	. . . . . . . . 9 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
126		ovex 7189	. . . . . . . . . . 11 ⊢ (𝑐 ⊕ 𝑎) ∈ V
127	126	mptex 6986	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) ∈ V
128	127	rnex 7617	. . . . . . . . 9 ⊢ ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) ∈ V
129	125, 50, 128	ovmpoa 7305	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝑋 ∧ (𝑐 ⊕ 𝑎) ∈ 𝑆) → (𝑏 ⊕ (𝑐 ⊕ 𝑎)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
130	101, 117, 129	syl2anc 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑏 ⊕ (𝑐 ⊕ 𝑎)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
131	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐺 ∈ Grp)
132	3, 9	grpcl 18111	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋) → (𝑏 + 𝑐) ∈ 𝑋)
133	131, 101, 84, 132	syl3anc 1367	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑏 + 𝑐) ∈ 𝑋)
134		simpr 487	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
135		simpl 485	. . . . . . . . . . . 12 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑥 = (𝑏 + 𝑐))
136	135	oveq1d 7171	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((𝑏 + 𝑐) + 𝑧))
137	134, 136	mpteq12dv 5151	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
138	137	rneqd 5808	. . . . . . . . 9 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
139	63	mptex 6986	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) ∈ V
140	139	rnex 7617	. . . . . . . . 9 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) ∈ V
141	138, 50, 140	ovmpoa 7305	. . . . . . . 8 ⊢ (((𝑏 + 𝑐) ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → ((𝑏 + 𝑐) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
142	133, 85, 141	syl2anc 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + 𝑐) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
143	115, 130, 142	3eqtr4rd 2867	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))
144	143	ralrimivva 3191	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))
145	80, 144	jca 514	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))
146	145	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))
147	52, 146	jca 514	. 2 ⊢ (𝜑 → ( ⊕ :(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑎 ∈ 𝑆 (((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))))
148	3, 9, 54	isga 18421	. 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))))))
149	7, 147, 148	sylanbrc 585	1 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ⊆ wss 3936  𝒫 cpw 4539   class class class wbr 5066   ↦ cmpt 5146   I cid 5459   × cxp 5553  ran crn 5556   ↾ cres 5557  ⟶wf 6351  –1-1→wf1 6352  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158   ≈ cen 8506  Fincfn 8509  ℕ₀cn0 11898  ↑cexp 13430  ♯chash 13691   ∥ cdvds 15607  ℙcprime 16015  Basecbs 16483  +_gcplusg 16565  0_gc0g 16713  Grpcgrp 18103   GrpAct cga 18419

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-hash 13692  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-ga 18420

This theorem is referenced by:  sylow1lem3  18725  sylow1lem5  18727