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Theorem sylow1lem2 19469

Description: Lemma for sylow1 19473. 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 6905	. . . . 5 ⊢ 𝑋 ∈ V
5	4	pwex 5378	. . . 4 ⊢ 𝒫 𝑋 ∈ V
6	2, 5	rabex2 5334	. . 3 ⊢ 𝑆 ∈ V
7	1, 6	jctir 521	. 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 2732	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧))
11	3, 9, 10	grplmulf1o 18899	. . . . . . . . . . . 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 6832	. . . . . . . . . . 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 6900	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑦 → ((♯‘𝑠) = (𝑃↑𝑁) ↔ (♯‘𝑦) = (𝑃↑𝑁)))
17	16, 2	elrab2 3686	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (♯‘𝑦) = (𝑃↑𝑁)))
18	15, 17	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑦 ∈ 𝒫 𝑋 ∧ (♯‘𝑦) = (𝑃↑𝑁)))
19	18	simpld 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝒫 𝑋)
20	19	elpwid 4611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ⊆ 𝑋)
21		f1ssres 6795	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦):𝑦–1-1→𝑋)
23		resmpt 6037	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑥 + 𝑧)) ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
24		f1eq1 6782	. . . . . . . . . 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 231	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋)
27		f1f 6787	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋 → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦⟶𝑋)
28		frn 6724	. . . . . . . 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 5345	. . . . . . 7 ⊢ (ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝒫 𝑋 ↔ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ⊆ 𝑋)
31	29, 30	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝒫 𝑋)
32		f1f1orn 6844	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1→𝑋 → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)):𝑦–1-1-onto→ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
33		vex 3478	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
34	33	f1oen 8971	. . . . . . . . 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 9175	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Fin ∧ 𝑦 ⊆ 𝑋) → 𝑦 ∈ Fin)
38	36, 20, 37	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ Fin)
39		ssfi 9175	. . . . . . . . . 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 14309	. . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ↔ 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
43	35, 42	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))))
44	18	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘𝑦) = (𝑃↑𝑁))
45	43, 44	eqtr3d 2774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) = (𝑃↑𝑁))
46		fveqeq2 6900	. . . . . . 7 ⊢ (𝑠 = ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) → ((♯‘𝑠) = (𝑃↑𝑁) ↔ (♯‘ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) = (𝑃↑𝑁)))
47	46, 2	elrab2 3686	. . . . . 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 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆)
49	48	ralrimivva 3200	. . . 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 8056	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ 𝑆 ↔ ⊕ :(𝑋 × 𝑆)⟶𝑆)
52	49, 51	sylib 217	. . 3 ⊢ (𝜑 → ⊕ :(𝑋 × 𝑆)⟶𝑆)
53	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐺 ∈ Grp)
54		eqid 2732	. . . . . . . . 9 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
55	3, 54	grpidcl 18852	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝑋)
56	53, 55	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0_g‘𝐺) ∈ 𝑋)
57		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
58		simpr 485	. . . . . . . . . 10 ⊢ ((𝑥 = (0_g‘𝐺) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
59		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑥 = (0_g‘𝐺) ∧ 𝑦 = 𝑎) → 𝑥 = (0_g‘𝐺))
60	59	oveq1d 7426	. . . . . . . . . 10 ⊢ ((𝑥 = (0_g‘𝐺) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((0_g‘𝐺) + 𝑧))
61	58, 60	mpteq12dv 5239	. . . . . . . . 9 ⊢ ((𝑥 = (0_g‘𝐺) ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)))
62	61	rneqd 5937	. . . . . . . 8 ⊢ ((𝑥 = (0_g‘𝐺) ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)))
63		vex 3478	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
64	63	mptex 7227	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)) ∈ V
65	64	rnex 7905	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)) ∈ V
66	62, 50, 65	ovmpoa 7565	. . . . . . 7 ⊢ (((0_g‘𝐺) ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → ((0_g‘𝐺) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)))
67	56, 57, 66	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0_g‘𝐺) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)))
68	2	ssrab3 4080	. . . . . . . . . . . . . 14 ⊢ 𝑆 ⊆ 𝒫 𝑋
69	68, 57	sselid 3980	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝒫 𝑋)
70	69	elpwid 4611	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ⊆ 𝑋)
71	70	sselda 3982	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑧 ∈ 𝑎) → 𝑧 ∈ 𝑋)
72	3, 9, 54	grplid 18854	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0_g‘𝐺) + 𝑧) = 𝑧)
73	53, 71, 72	syl2an2r 683	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑧 ∈ 𝑎) → ((0_g‘𝐺) + 𝑧) = 𝑧)
74	73	mpteq2dva 5248	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)) = (𝑧 ∈ 𝑎 ↦ 𝑧))
75		mptresid 6050	. . . . . . . . 9 ⊢ ( I ↾ 𝑎) = (𝑧 ∈ 𝑎 ↦ 𝑧)
76	74, 75	eqtr4di 2790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)) = ( I ↾ 𝑎))
77	76	rneqd 5937	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ran (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)) = ran ( I ↾ 𝑎))
78		rnresi 6074	. . . . . . 7 ⊢ ran ( I ↾ 𝑎) = 𝑎
79	77, 78	eqtrdi 2788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ran (𝑧 ∈ 𝑎 ↦ ((0_g‘𝐺) + 𝑧)) = 𝑎)
80	67, 79	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0_g‘𝐺) ⊕ 𝑎) = 𝑎)
81		ovex 7444	. . . . . . . . . 10 ⊢ (𝑐 + 𝑧) ∈ V
82		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑤 = (𝑐 + 𝑧) → (𝑏 + 𝑤) = (𝑏 + (𝑐 + 𝑧)))
83	81, 82	abrexco 7245	. . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑎 ∈ 𝑆)
86		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
87		simpl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → 𝑥 = 𝑐)
88	87	oveq1d 7426	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = (𝑐 + 𝑧))
89	86, 88	mpteq12dv 5239	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
90	89	rneqd 5937	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
91	63	mptex 7227	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) ∈ V
92	91	rnex 7905	. . . . . . . . . . . . . 14 ⊢ ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) ∈ V
93	90, 50, 92	ovmpoa 7565	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → (𝑐 ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
94	84, 85, 93	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)))
95		eqid 2732	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧))
96	95	rnmpt 5954	. . . . . . . . . . . 12 ⊢ ran (𝑧 ∈ 𝑎 ↦ (𝑐 + 𝑧)) = {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)}
97	94, 96	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) = {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = (𝑐 + 𝑧)})
98	97	rexeqdv 3326	. . . . . . . . . 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 2801	. . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑏 ∈ 𝑋)
103	84	adantr 481	. . . . . . . . . . . . 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 18830	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧)))
106	100, 102, 103, 104, 105	syl13anc 1372	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧)))
107	106	eqeq2d 2743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (𝑢 = ((𝑏 + 𝑐) + 𝑧) ↔ 𝑢 = (𝑏 + (𝑐 + 𝑧))))
108	107	rexbidva 3176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧) ↔ ∃𝑧 ∈ 𝑎 𝑢 = (𝑏 + (𝑐 + 𝑧))))
109	108	abbidv 2801	. . . . . . . . 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 2798	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧)})
111		eqid 2732	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤))
112	111	rnmpt 5954	. . . . . . . 8 ⊢ ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) = {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = (𝑏 + 𝑤)}
113		eqid 2732	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧))
114	113	rnmpt 5954	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + 𝑐) + 𝑧)}
115	110, 112, 114	3eqtr4g 2797	. . . . . . 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	fovcdmd 7581	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) ∈ 𝑆)
118		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → 𝑦 = (𝑐 ⊕ 𝑎))
119		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → 𝑥 = 𝑏)
120	119	oveq1d 7426	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑥 + 𝑧) = (𝑏 + 𝑧))
121	118, 120	mpteq12dv 5239	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑧)))
122		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑏 + 𝑧) = (𝑏 + 𝑤))
123	122	cbvmptv 5261	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑧)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤))
124	121, 123	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
125	124	rneqd 5937	. . . . . . . . 9 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
126		ovex 7444	. . . . . . . . . . 11 ⊢ (𝑐 ⊕ 𝑎) ∈ V
127	126	mptex 7227	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) ∈ V
128	127	rnex 7905	. . . . . . . . 9 ⊢ ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)) ∈ V
129	125, 50, 128	ovmpoa 7565	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝑋 ∧ (𝑐 ⊕ 𝑎) ∈ 𝑆) → (𝑏 ⊕ (𝑐 ⊕ 𝑎)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ (𝑏 + 𝑤)))
130	101, 117, 129	syl2anc 584	. . . . . . 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 18829	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋) → (𝑏 + 𝑐) ∈ 𝑋)
133	131, 101, 84, 132	syl3anc 1371	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑏 + 𝑐) ∈ 𝑋)
134		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
135		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑥 = (𝑏 + 𝑐))
136	135	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((𝑏 + 𝑐) + 𝑧))
137	134, 136	mpteq12dv 5239	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
138	137	rneqd 5937	. . . . . . . . 9 ⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
139	63	mptex 7227	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) ∈ V
140	139	rnex 7905	. . . . . . . . 9 ⊢ ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)) ∈ V
141	138, 50, 140	ovmpoa 7565	. . . . . . . 8 ⊢ (((𝑏 + 𝑐) ∈ 𝑋 ∧ 𝑎 ∈ 𝑆) → ((𝑏 + 𝑐) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
142	133, 85, 141	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + 𝑐) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((𝑏 + 𝑐) + 𝑧)))
143	115, 130, 142	3eqtr4rd 2783	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))
144	143	ralrimivva 3200	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))
145	80, 144	jca 512	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((0_g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))
146	145	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (((0_g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))
147	52, 146	jca 512	. 2 ⊢ (𝜑 → ( ⊕ :(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑎 ∈ 𝑆 (((0_g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))))
148	3, 9, 54	isga 19157	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) ↔ ((𝐺 ∈ Grp ∧ 𝑆 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑎 ∈ 𝑆 (((0_g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))))
149	7, 147, 148	sylanbrc 583	1 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ⊆ wss 3948 𝒫 cpw 4602 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 × cxp 5674 ran crn 5677 ↾ cres 5678 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 ≈ cen 8938 Fincfn 8941 ℕ₀cn0 12474 ↑cexp 14029 ♯chash 14292 ∥ cdvds 16199 ℙcprime 16610 Basecbs 17146 +_gcplusg 17199 0_gc0g 17387 Grpcgrp 18821 GrpAct cga 19155

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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-n0 12475 df-z 12561 df-uz 12825 df-hash 14293 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-ga 19156

This theorem is referenced by: sylow1lem3 19470 sylow1lem5 19472

W3C validator