Theorem odcau 18019

Description: Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime 𝑃 contains an element of order 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)

Hypotheses

Ref	Expression
odcau.x	⊢ 𝑋 = (Base‘𝐺)
odcau.o	⊢ 𝑂 = (od‘𝐺)

Assertion

Ref	Expression
odcau	⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃)

Distinct variable groups:   𝑔,𝐺   𝑃,𝑔   𝑔,𝑋

Allowed substitution hint:   𝑂(𝑔)

Proof of Theorem odcau

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		odcau.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		simpl1 1064	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝐺 ∈ Grp)
3		simpl2 1065	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑋 ∈ Fin)
4		simpl3 1066	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℙ)
5		1nn0 11308	. . . 4 ⊢ 1 ∈ ℕ0
6	5	a1i 11	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 1 ∈ ℕ0)
7		prmnn 15388	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8	4, 7	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℕ)
9	8	nncnd 11036	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℂ)
10	9	exp1d 13003	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → (𝑃↑1) = 𝑃)
11		simpr 477	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∥ (#‘𝑋))
12	10, 11	eqbrtrd 4675	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → (𝑃↑1) ∥ (#‘𝑋))
13	1, 2, 3, 4, 6, 12	sylow1 18018	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ∃𝑠 ∈ (SubGrp‘𝐺)(#‘𝑠) = (𝑃↑1))
14	10	eqeq2d 2632	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ((#‘𝑠) = (𝑃↑1) ↔ (#‘𝑠) = 𝑃))
15	14	adantr 481	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ 𝑠 ∈ (SubGrp‘𝐺)) → ((#‘𝑠) = (𝑃↑1) ↔ (#‘𝑠) = 𝑃))
16		fvex 6201	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) ∈ V
17		hashsng 13159	. . . . . . . . . . . 12 ⊢ ((0g‘𝐺) ∈ V → (#‘{(0g‘𝐺)}) = 1)
18	16, 17	ax-mp 5	. . . . . . . . . . 11 ⊢ (#‘{(0g‘𝐺)}) = 1
19		simprr 796	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘𝑠) = 𝑃)
20	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑃 ∈ ℙ)
21		prmuz2 15408	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
22	20, 21	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑃 ∈ (ℤ≥‘2))
23	19, 22	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘𝑠) ∈ (ℤ≥‘2))
24		eluz2b2 11761	. . . . . . . . . . . . 13 ⊢ ((#‘𝑠) ∈ (ℤ≥‘2) ↔ ((#‘𝑠) ∈ ℕ ∧ 1 < (#‘𝑠)))
25	24	simprbi 480	. . . . . . . . . . . 12 ⊢ ((#‘𝑠) ∈ (ℤ≥‘2) → 1 < (#‘𝑠))
26	23, 25	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 1 < (#‘𝑠))
27	18, 26	syl5eqbr 4688	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘{(0g‘𝐺)}) < (#‘𝑠))
28		snfi 8038	. . . . . . . . . . 11 ⊢ {(0g‘𝐺)} ∈ Fin
29	3	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑋 ∈ Fin)
30	1	subgss 17595	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ⊆ 𝑋)
31	30	ad2antrl 764	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑠 ⊆ 𝑋)
32		ssfi 8180	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Fin ∧ 𝑠 ⊆ 𝑋) → 𝑠 ∈ Fin)
33	29, 31, 32	syl2anc 693	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑠 ∈ Fin)
34		hashsdom 13170	. . . . . . . . . . 11 ⊢ (({(0g‘𝐺)} ∈ Fin ∧ 𝑠 ∈ Fin) → ((#‘{(0g‘𝐺)}) < (#‘𝑠) ↔ {(0g‘𝐺)} ≺ 𝑠))
35	28, 33, 34	sylancr 695	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ((#‘{(0g‘𝐺)}) < (#‘𝑠) ↔ {(0g‘𝐺)} ≺ 𝑠))
36	27, 35	mpbid 222	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → {(0g‘𝐺)} ≺ 𝑠)
37		sdomdif 8108	. . . . . . . . 9 ⊢ ({(0g‘𝐺)} ≺ 𝑠 → (𝑠 ∖ {(0g‘𝐺)}) ≠ ∅)
38	36, 37	syl 17	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (𝑠 ∖ {(0g‘𝐺)}) ≠ ∅)
39		n0 3931	. . . . . . . 8 ⊢ ((𝑠 ∖ {(0g‘𝐺)}) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑠 ∖ {(0g‘𝐺)}))
40	38, 39	sylib 208	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ∃𝑔 𝑔 ∈ (𝑠 ∖ {(0g‘𝐺)}))
41		eldifsn 4317	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝑠 ∖ {(0g‘𝐺)}) ↔ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))
42	31	adantrr 753	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝑠 ⊆ 𝑋)
43		simprrl 804	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝑔 ∈ 𝑠)
44	42, 43	sseldd 3604	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝑔 ∈ 𝑋)
45		simprrr 805	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝑔 ≠ (0g‘𝐺))
46		simprll 802	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝑠 ∈ (SubGrp‘𝐺))
47	33	adantrr 753	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝑠 ∈ Fin)
48		odcau.o	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑂 = (od‘𝐺)
49	48	odsubdvds 17986	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ Fin ∧ 𝑔 ∈ 𝑠) → (𝑂‘𝑔) ∥ (#‘𝑠))
50	46, 47, 43, 49	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (𝑂‘𝑔) ∥ (#‘𝑠))
51		simprlr 803	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (#‘𝑠) = 𝑃)
52	50, 51	breqtrd 4679	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (𝑂‘𝑔) ∥ 𝑃)
53	4	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝑃 ∈ ℙ)
54	2	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝐺 ∈ Grp)
55	3	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → 𝑋 ∈ Fin)
56	1, 48	odcl2 17982	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑔 ∈ 𝑋) → (𝑂‘𝑔) ∈ ℕ)
57	54, 55, 44, 56	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (𝑂‘𝑔) ∈ ℕ)
58		dvdsprime 15400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℙ ∧ (𝑂‘𝑔) ∈ ℕ) → ((𝑂‘𝑔) ∥ 𝑃 ↔ ((𝑂‘𝑔) = 𝑃 ∨ (𝑂‘𝑔) = 1)))
59	53, 57, 58	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → ((𝑂‘𝑔) ∥ 𝑃 ↔ ((𝑂‘𝑔) = 𝑃 ∨ (𝑂‘𝑔) = 1)))
60	52, 59	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → ((𝑂‘𝑔) = 𝑃 ∨ (𝑂‘𝑔) = 1))
61	60	ord 392	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (¬ (𝑂‘𝑔) = 𝑃 → (𝑂‘𝑔) = 1))
62		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐺) = (0g‘𝐺)
63	48, 62, 1	odeq1 17977	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ 𝑔 ∈ 𝑋) → ((𝑂‘𝑔) = 1 ↔ 𝑔 = (0g‘𝐺)))
64	54, 44, 63	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → ((𝑂‘𝑔) = 1 ↔ 𝑔 = (0g‘𝐺)))
65	61, 64	sylibd 229	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (¬ (𝑂‘𝑔) = 𝑃 → 𝑔 = (0g‘𝐺)))
66	65	necon1ad 2811	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (𝑔 ≠ (0g‘𝐺) → (𝑂‘𝑔) = 𝑃))
67	45, 66	mpd 15	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (𝑂‘𝑔) = 𝑃)
68	44, 67	jca 554	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)))) → (𝑔 ∈ 𝑋 ∧ (𝑂‘𝑔) = 𝑃))
69	68	expr 643	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ((𝑔 ∈ 𝑠 ∧ 𝑔 ≠ (0g‘𝐺)) → (𝑔 ∈ 𝑋 ∧ (𝑂‘𝑔) = 𝑃)))
70	41, 69	syl5bi 232	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (𝑔 ∈ (𝑠 ∖ {(0g‘𝐺)}) → (𝑔 ∈ 𝑋 ∧ (𝑂‘𝑔) = 𝑃)))
71	70	eximdv 1846	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (∃𝑔 𝑔 ∈ (𝑠 ∖ {(0g‘𝐺)}) → ∃𝑔(𝑔 ∈ 𝑋 ∧ (𝑂‘𝑔) = 𝑃)))
72	40, 71	mpd 15	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ∃𝑔(𝑔 ∈ 𝑋 ∧ (𝑂‘𝑔) = 𝑃))
73		df-rex 2918	. . . . . 6 ⊢ (∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃 ↔ ∃𝑔(𝑔 ∈ 𝑋 ∧ (𝑂‘𝑔) = 𝑃))
74	72, 73	sylibr 224	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃)
75	74	expr 643	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ 𝑠 ∈ (SubGrp‘𝐺)) → ((#‘𝑠) = 𝑃 → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃))
76	15, 75	sylbid 230	. . 3 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ 𝑠 ∈ (SubGrp‘𝐺)) → ((#‘𝑠) = (𝑃↑1) → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃))
77	76	rexlimdva 3031	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → (∃𝑠 ∈ (SubGrp‘𝐺)(#‘𝑠) = (𝑃↑1) → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃))
78	13, 77	mpd 15	1 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650   ≺ csdm 7954  Fincfn 7955  1c1 9937   < clt 10074  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤ_≥cuz 11687  ↑cexp 12860  #chash 13117   ∥ cdvds 14983  ℙcprime 15385  Basecbs 15857  0_gc0g 16100  Grpcgrp 17422  SubGrpcsubg 17588  odcod 17944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-eqg 17593  df-ga 17723  df-od 17948

This theorem is referenced by:  pgpfi  18020  ablfacrplem  18464