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Theorem odcau 17940
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 1062 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝐺 ∈ Grp)
3 simpl2 1063 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑋 ∈ Fin)
4 simpl3 1064 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℙ)
5 1nn0 11252 . . . 4 1 ∈ ℕ0
65a1i 11 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 1 ∈ ℕ0)
7 prmnn 15312 . . . . . . 7 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
84, 7syl 17 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℕ)
98nncnd 10980 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → 𝑃 ∈ ℂ)
109exp1d 12943 . . . 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))
1210, 11eqbrtrd 4635 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → (𝑃↑1) ∥ (#‘𝑋))
131, 2, 3, 4, 6, 12sylow1 17939 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ∃𝑠 ∈ (SubGrp‘𝐺)(#‘𝑠) = (𝑃↑1))
1410eqeq2d 2631 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ((#‘𝑠) = (𝑃↑1) ↔ (#‘𝑠) = 𝑃))
1514adantr 481 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ 𝑠 ∈ (SubGrp‘𝐺)) → ((#‘𝑠) = (𝑃↑1) ↔ (#‘𝑠) = 𝑃))
16 fvex 6158 . . . . . . . . . . . 12 (0g𝐺) ∈ V
17 hashsng 13099 . . . . . . . . . . . 12 ((0g𝐺) ∈ V → (#‘{(0g𝐺)}) = 1)
1816, 17ax-mp 5 . . . . . . . . . . 11 (#‘{(0g𝐺)}) = 1
19 simprr 795 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘𝑠) = 𝑃)
204adantr 481 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑃 ∈ ℙ)
21 prmuz2 15332 . . . . . . . . . . . . . 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 2698 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘𝑠) ∈ (ℤ‘2))
24 eluz2b2 11705 . . . . . . . . . . . . 13 ((#‘𝑠) ∈ (ℤ‘2) ↔ ((#‘𝑠) ∈ ℕ ∧ 1 < (#‘𝑠)))
2524simprbi 480 . . . . . . . . . . . 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 4648 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → (#‘{(0g𝐺)}) < (#‘𝑠))
28 snfi 7982 . . . . . . . . . . 11 {(0g𝐺)} ∈ Fin
293adantr 481 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑋 ∈ Fin)
301subgss 17516 . . . . . . . . . . . . 13 (𝑠 ∈ (SubGrp‘𝐺) → 𝑠𝑋)
3130ad2antrl 763 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑠𝑋)
32 ssfi 8124 . . . . . . . . . . . 12 ((𝑋 ∈ Fin ∧ 𝑠𝑋) → 𝑠 ∈ Fin)
3329, 31, 32syl2anc 692 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃)) → 𝑠 ∈ Fin)
34 hashsdom 13110 . . . . . . . . . . 11 (({(0g𝐺)} ∈ Fin ∧ 𝑠 ∈ Fin) → ((#‘{(0g𝐺)}) < (#‘𝑠) ↔ {(0g𝐺)} ≺ 𝑠))
3528, 33, 34sylancr 694 . . . . . . . . . 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 8052 . . . . . . . . 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 3907 . . . . . . . 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 4287 . . . . . . . . 9 (𝑔 ∈ (𝑠 ∖ {(0g𝐺)}) ↔ (𝑔𝑠𝑔 ≠ (0g𝐺)))
4231adantrr 752 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑠𝑋)
43 simprrl 803 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑔𝑠)
4442, 43sseldd 3584 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑔𝑋)
45 simprrr 804 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑔 ≠ (0g𝐺))
46 simprll 801 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑠 ∈ (SubGrp‘𝐺))
4733adantrr 752 . . . . . . . . . . . . . . . . . 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 17907 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ Fin ∧ 𝑔𝑠) → (𝑂𝑔) ∥ (#‘𝑠))
5046, 47, 43, 49syl3anc 1323 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑂𝑔) ∥ (#‘𝑠))
51 simprlr 802 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (#‘𝑠) = 𝑃)
5250, 51breqtrd 4639 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑂𝑔) ∥ 𝑃)
534adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑃 ∈ ℙ)
542adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝐺 ∈ Grp)
553adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → 𝑋 ∈ Fin)
561, 48odcl2 17903 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑔𝑋) → (𝑂𝑔) ∈ ℕ)
5754, 55, 44, 56syl3anc 1323 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (𝑂𝑔) ∈ ℕ)
58 dvdsprime 15324 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (𝑂𝑔) ∈ ℕ) → ((𝑂𝑔) ∥ 𝑃 ↔ ((𝑂𝑔) = 𝑃 ∨ (𝑂𝑔) = 1)))
5953, 57, 58syl2anc 692 . . . . . . . . . . . . . . . 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 392 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) ∧ ((𝑠 ∈ (SubGrp‘𝐺) ∧ (#‘𝑠) = 𝑃) ∧ (𝑔𝑠𝑔 ≠ (0g𝐺)))) → (¬ (𝑂𝑔) = 𝑃 → (𝑂𝑔) = 1))
62 eqid 2621 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
6348, 62, 1odeq1 17898 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑔𝑋) → ((𝑂𝑔) = 1 ↔ 𝑔 = (0g𝐺)))
6454, 44, 63syl2anc 692 . . . . . . . . . . . . . 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 2807 . . . . . . . . . . . 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 554 . . . . . . . . . 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 1843 . . . . . . 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 2913 . . . . . 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 3024 . 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 383  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  c0 3891  {csn 4148   class class class wbr 4613  cfv 5847  (class class class)co 6604  csdm 7898  Fincfn 7899  1c1 9881   < clt 10018  cn 10964  2c2 11014  0cn0 11236  cuz 11631  cexp 12800  #chash 13057  cdvds 14907  cprime 15309  Basecbs 15781  0gc0g 16021  Grpcgrp 17343  SubGrpcsubg 17509  odcod 17865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-acn 8712  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-fac 13001  df-bc 13030  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-dvds 14908  df-gcd 15141  df-prm 15310  df-pc 15466  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-eqg 17514  df-ga 17644  df-od 17869
This theorem is referenced by:  pgpfi  17941  ablfacrplem  18385
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