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Theorem odcau 18065
 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)) → ∃𝑔𝑋 (𝑂𝑔) = 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.
StepHypRef Expression
1 odcau.x . . 3 𝑋 = (Base‘𝐺)
2 simpl1 1084 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝐺 ∈ Grp)
3 simpl2 1085 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑋 ∈ Fin)
4 simpl3 1086 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℙ)
5 1nn0 11346 . . . 4 1 ∈ ℕ0
65a1i 11 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 1 ∈ ℕ0)
7 prmnn 15435 . . . . . . 7 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
84, 7syl 17 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℕ)
98nncnd 11074 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℂ)
109exp1d 13043 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → (𝑃↑1) = 𝑃)
11 simpr 476 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∥ (#‘𝑋))
1210, 11eqbrtrd 4707 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → (𝑃↑1) ∥ (#‘𝑋))
131, 2, 3, 4, 6, 12sylow1 18064 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ∃𝑠 ∈ (SubGrp‘𝐺)(#‘𝑠) = (𝑃↑1))
1410eqeq2d 2661 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ((#‘𝑠) = (𝑃↑1) ↔ (#‘𝑠) = 𝑃))
1514adantr 480 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ 𝑠 ∈ (SubGrp‘𝐺)) → ((#‘𝑠) = (𝑃↑1) ↔ (#‘𝑠) = 𝑃))
16 fvex 6239 . . . . . . . . . . . 12 (0g𝐺) ∈ V
17 hashsng 13197 . . . . . . . . . . . 12 ((0g𝐺) ∈ V → (#‘{(0g𝐺)}) = 1)
1816, 17ax-mp 5 . . . . . . . . . . 11 (#‘{(0g𝐺)}) = 1
19 simprr 811 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘𝑠) = 𝑃)
204adantr 480 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑃 ∈ ℙ)
21 prmuz2 15455 . . . . . . . . . . . . . 14 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
2220, 21syl 17 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑃 ∈ (ℤ‘2))
2319, 22eqeltrd 2730 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘𝑠) ∈ (ℤ‘2))
24 eluz2b2 11799 . . . . . . . . . . . . 13 ((#‘𝑠) ∈ (ℤ‘2) ↔ ((#‘𝑠) ∈ ℕ ∧ 1 < (#‘𝑠)))
2524simprbi 479 . . . . . . . . . . . 12 ((#‘𝑠) ∈ (ℤ‘2) → 1 < (#‘𝑠))
2623, 25syl 17 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 1 < (#‘𝑠))
2718, 26syl5eqbr 4720 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘{(0g𝐺)}) < (#‘𝑠))
28 snfi 8079 . . . . . . . . . . 11 {(0g𝐺)} ∈ Fin
293adantr 480 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑋 ∈ Fin)
301subgss 17642 . . . . . . . . . . . . 13 (𝑠 ∈ (SubGrp‘𝐺) → 𝑠𝑋)
3130ad2antrl 764 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑠𝑋)
32 ssfi 8221 . . . . . . . . . . . 12 ((𝑋 ∈ Fin ∧ 𝑠𝑋) → 𝑠 ∈ Fin)
3329, 31, 32syl2anc 694 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑠 ∈ Fin)
34 hashsdom 13208 . . . . . . . . . . 11 (({(0g𝐺)} ∈ Fin ∧ 𝑠 ∈ Fin) → ((#‘{(0g𝐺)}) < (#‘𝑠) ↔ {(0g𝐺)} ≺ 𝑠))
3528, 33, 34sylancr 696 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ((#‘{(0g𝐺)}) < (#‘𝑠) ↔ {(0g𝐺)} ≺ 𝑠))
3627, 35mpbid 222 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → {(0g𝐺)} ≺ 𝑠)
37 sdomdif 8149 . . . . . . . . 9 ({(0g𝐺)} ≺ 𝑠 → (𝑠 ∖ {(0g𝐺)}) ≠ ∅)
3836, 37syl 17 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (𝑠 ∖ {(0g𝐺)}) ≠ ∅)
39 n0 3964 . . . . . . . 8 ((𝑠 ∖ {(0g𝐺)}) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑠 ∖ {(0g𝐺)}))
4038, 39sylib 208 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ∃𝑔 𝑔 ∈ (𝑠 ∖ {(0g𝐺)}))
41 eldifsn 4350 . . . . . . . . 9 (𝑔 ∈ (𝑠 ∖ {(0g𝐺)}) ↔ (𝑔𝑠𝑔 ≠ (0g𝐺)))
4231adantrr 753 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑠𝑋)
43 simprrl 821 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑔𝑠)
4442, 43sseldd 3637 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑔𝑋)
45 simprrr 822 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑔 ≠ (0g𝐺))
46 simprll 819 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑠 ∈ (SubGrp‘𝐺))
4733adantrr 753 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑠 ∈ Fin)
48 odcau.o . . . . . . . . . . . . . . . . . . 19 𝑂 = (od‘𝐺)
4948odsubdvds 18032 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ Fin ∧ 𝑔𝑠) → (𝑂𝑔) ∥ (#‘𝑠))
5046, 47, 43, 49syl3anc 1366 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑂𝑔) ∥ (#‘𝑠))
51 simprlr 820 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (#‘𝑠) = 𝑃)
5250, 51breqtrd 4711 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑂𝑔) ∥ 𝑃)
534adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑃 ∈ ℙ)
542adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝐺 ∈ Grp)
553adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑋 ∈ Fin)
561, 48odcl2 18028 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑔𝑋) → (𝑂𝑔) ∈ ℕ)
5754, 55, 44, 56syl3anc 1366 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑂𝑔) ∈ ℕ)
58 dvdsprime 15447 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (𝑂𝑔) ∈ ℕ) → ((𝑂𝑔) ∥ 𝑃 ↔ ((𝑂𝑔) = 𝑃 ∨ (𝑂𝑔) = 1)))
5953, 57, 58syl2anc 694 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → ((𝑂𝑔) ∥ 𝑃 ↔ ((𝑂𝑔) = 𝑃 ∨ (𝑂𝑔) = 1)))
6052, 59mpbid 222 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → ((𝑂𝑔) = 𝑃 ∨ (𝑂𝑔) = 1))
6160ord 391 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (¬ (𝑂𝑔) = 𝑃 → (𝑂𝑔) = 1))
62 eqid 2651 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
6348, 62, 1odeq1 18023 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑔𝑋) → ((𝑂𝑔) = 1 ↔ 𝑔 = (0g𝐺)))
6454, 44, 63syl2anc 694 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → ((𝑂𝑔) = 1 ↔ 𝑔 = (0g𝐺)))
6561, 64sylibd 229 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (¬ (𝑂𝑔) = 𝑃𝑔 = (0g𝐺)))
6665necon1ad 2840 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑔 ≠ (0g𝐺) → (𝑂𝑔) = 𝑃))
6745, 66mpd 15 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑂𝑔) = 𝑃)
6844, 67jca 553 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑔𝑋 ∧ (𝑂𝑔) = 𝑃))
6968expr 642 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ((𝑔𝑠𝑔 ≠ (0g𝐺)) → (𝑔𝑋 ∧ (𝑂𝑔) = 𝑃)))
7041, 69syl5bi 232 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (𝑔 ∈ (𝑠 ∖ {(0g𝐺)}) → (𝑔𝑋 ∧ (𝑂𝑔) = 𝑃)))
7170eximdv 1886 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (∃𝑔 𝑔 ∈ (𝑠 ∖ {(0g𝐺)}) → ∃𝑔(𝑔𝑋 ∧ (𝑂𝑔) = 𝑃)))
7240, 71mpd 15 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ∃𝑔(𝑔𝑋 ∧ (𝑂𝑔) = 𝑃))
73 df-rex 2947 . . . . . 6 (∃𝑔𝑋 (𝑂𝑔) = 𝑃 ↔ ∃𝑔(𝑔𝑋 ∧ (𝑂𝑔) = 𝑃))
7472, 73sylibr 224 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → ∃𝑔𝑋 (𝑂𝑔) = 𝑃)
7574expr 642 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ 𝑠 ∈ (SubGrp‘𝐺)) → ((#‘𝑠) = 𝑃 → ∃𝑔𝑋 (𝑂𝑔) = 𝑃))
7615, 75sylbid 230 . . 3 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ 𝑠 ∈ (SubGrp‘𝐺)) → ((#‘𝑠) = (𝑃↑1) → ∃𝑔𝑋 (𝑂𝑔) = 𝑃))
7776rexlimdva 3060 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → (∃𝑠 ∈ (SubGrp‘𝐺)(#‘𝑠) = (𝑃↑1) → ∃𝑔𝑋 (𝑂𝑔) = 𝑃))
7813, 77mpd 15 1 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ∃𝑔𝑋 (𝑂𝑔) = 𝑃)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823  ∃wrex 2942  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  ∅c0 3948  {csn 4210   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690   ≺ csdm 7996  Fincfn 7997  1c1 9975   < clt 10112  ℕcn 11058  2c2 11108  ℕ0cn0 11330  ℤ≥cuz 11725  ↑cexp 12900  #chash 13157   ∥ cdvds 15027  ℙcprime 15432  Basecbs 15904  0gc0g 16147  Grpcgrp 17469  SubGrpcsubg 17635  odcod 17990 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-eqg 17640  df-ga 17769  df-od 17994 This theorem is referenced by:  pgpfi  18066  ablfacrplem  18510
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