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Theorem sylow3lem1 18754
 Description: Lemma for sylow3 18760, first part. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypotheses
Ref Expression
sylow3.x 𝑋 = (Base‘𝐺)
sylow3.g (𝜑𝐺 ∈ Grp)
sylow3.xf (𝜑𝑋 ∈ Fin)
sylow3.p (𝜑𝑃 ∈ ℙ)
sylow3lem1.a + = (+g𝐺)
sylow3lem1.d = (-g𝐺)
sylow3lem1.m = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))
Assertion
Ref Expression
sylow3lem1 (𝜑 ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)))
Distinct variable groups:   𝑥,𝑦,𝑧,   𝑥, ,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥,𝑃,𝑦,𝑧

Proof of Theorem sylow3lem1
Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow3.g . . 3 (𝜑𝐺 ∈ Grp)
2 ovex 7184 . . 3 (𝑃 pSyl 𝐺) ∈ V
31, 2jctir 524 . 2 (𝜑 → (𝐺 ∈ Grp ∧ (𝑃 pSyl 𝐺) ∈ V))
4 sylow3.xf . . . . . . . . . . 11 (𝜑𝑋 ∈ Fin)
5 sylow3.p . . . . . . . . . . 11 (𝜑𝑃 ∈ ℙ)
6 sylow3.x . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
76fislw 18752 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝑦 ∈ (𝑃 pSyl 𝐺) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))
81, 4, 5, 7syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝑦 ∈ (𝑃 pSyl 𝐺) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))
98biimpa 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑃 pSyl 𝐺)) → (𝑦 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
109adantrl 715 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → (𝑦 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
1110simpld 498 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑦 ∈ (SubGrp‘𝐺))
12 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑥𝑋)
13 sylow3lem1.a . . . . . . . 8 + = (+g𝐺)
14 sylow3lem1.d . . . . . . . 8 = (-g𝐺)
15 eqid 2824 . . . . . . . 8 (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))
166, 13, 14, 15conjsubg 18392 . . . . . . 7 ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (SubGrp‘𝐺))
1711, 12, 16syl2anc 587 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (SubGrp‘𝐺))
186, 13, 14, 15conjsubgen 18393 . . . . . . . . 9 ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑦 ≈ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))
1911, 12, 18syl2anc 587 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑦 ≈ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))
204adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑋 ∈ Fin)
216subgss 18282 . . . . . . . . . . 11 (𝑦 ∈ (SubGrp‘𝐺) → 𝑦𝑋)
2211, 21syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑦𝑋)
2320, 22ssfid 8740 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑦 ∈ Fin)
246subgss 18282 . . . . . . . . . . 11 (ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (SubGrp‘𝐺) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ⊆ 𝑋)
2517, 24syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ⊆ 𝑋)
2620, 25ssfid 8740 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ Fin)
27 hashen 13714 . . . . . . . . 9 ((𝑦 ∈ Fin ∧ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ Fin) → ((♯‘𝑦) = (♯‘ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))) ↔ 𝑦 ≈ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))))
2823, 26, 27syl2anc 587 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → ((♯‘𝑦) = (♯‘ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))) ↔ 𝑦 ≈ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))))
2919, 28mpbird 260 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → (♯‘𝑦) = (♯‘ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))))
3010simprd 499 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
3129, 30eqtr3d 2861 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → (♯‘ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
326fislw 18752 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (𝑃 pSyl 𝐺) ↔ (ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (SubGrp‘𝐺) ∧ (♯‘ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))
331, 4, 5, 32syl3anc 1368 . . . . . . 7 (𝜑 → (ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (𝑃 pSyl 𝐺) ↔ (ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (SubGrp‘𝐺) ∧ (♯‘ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))
3433adantr 484 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → (ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (𝑃 pSyl 𝐺) ↔ (ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (SubGrp‘𝐺) ∧ (♯‘ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))
3517, 31, 34mpbir2and 712 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺))) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (𝑃 pSyl 𝐺))
3635ralrimivva 3186 . . . 4 (𝜑 → ∀𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺)ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (𝑃 pSyl 𝐺))
37 sylow3lem1.m . . . . 5 = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))
3837fmpo 7763 . . . 4 (∀𝑥𝑋𝑦 ∈ (𝑃 pSyl 𝐺)ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) ∈ (𝑃 pSyl 𝐺) ↔ :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺))
3936, 38sylib 221 . . 3 (𝜑 :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺))
401adantr 484 . . . . . . . 8 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → 𝐺 ∈ Grp)
41 eqid 2824 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
426, 41grpidcl 18133 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
4340, 42syl 17 . . . . . . 7 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → (0g𝐺) ∈ 𝑋)
44 simpr 488 . . . . . . 7 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → 𝑎 ∈ (𝑃 pSyl 𝐺))
45 simpr 488 . . . . . . . . . 10 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
46 simpl 486 . . . . . . . . . . . 12 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑎) → 𝑥 = (0g𝐺))
4746oveq1d 7166 . . . . . . . . . . 11 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((0g𝐺) + 𝑧))
4847, 46oveq12d 7169 . . . . . . . . . 10 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑎) → ((𝑥 + 𝑧) 𝑥) = (((0g𝐺) + 𝑧) (0g𝐺)))
4945, 48mpteq12dv 5138 . . . . . . . . 9 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑎) → (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))))
5049rneqd 5796 . . . . . . . 8 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑎) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = ran (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))))
51 vex 3483 . . . . . . . . . 10 𝑎 ∈ V
5251mptex 6979 . . . . . . . . 9 (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))) ∈ V
5352rnex 7614 . . . . . . . 8 ran (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))) ∈ V
5450, 37, 53ovmpoa 7300 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑎 ∈ (𝑃 pSyl 𝐺)) → ((0g𝐺) 𝑎) = ran (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))))
5543, 44, 54syl2anc 587 . . . . . 6 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → ((0g𝐺) 𝑎) = ran (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))))
561ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧𝑎) → 𝐺 ∈ Grp)
57 slwsubg 18737 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (𝑃 pSyl 𝐺) → 𝑎 ∈ (SubGrp‘𝐺))
5857adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → 𝑎 ∈ (SubGrp‘𝐺))
596subgss 18282 . . . . . . . . . . . . . . 15 (𝑎 ∈ (SubGrp‘𝐺) → 𝑎𝑋)
6058, 59syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → 𝑎𝑋)
6160sselda 3953 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧𝑎) → 𝑧𝑋)
626, 13, 41grplid 18135 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((0g𝐺) + 𝑧) = 𝑧)
6356, 61, 62syl2anc 587 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧𝑎) → ((0g𝐺) + 𝑧) = 𝑧)
6463oveq1d 7166 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧𝑎) → (((0g𝐺) + 𝑧) (0g𝐺)) = (𝑧 (0g𝐺)))
656, 41, 14grpsubid1 18186 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → (𝑧 (0g𝐺)) = 𝑧)
6656, 61, 65syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧𝑎) → (𝑧 (0g𝐺)) = 𝑧)
6764, 66eqtrd 2859 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧𝑎) → (((0g𝐺) + 𝑧) (0g𝐺)) = 𝑧)
6867mpteq2dva 5148 . . . . . . . . 9 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))) = (𝑧𝑎𝑧))
69 mptresid 5907 . . . . . . . . 9 ( I ↾ 𝑎) = (𝑧𝑎𝑧)
7068, 69eqtr4di 2877 . . . . . . . 8 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))) = ( I ↾ 𝑎))
7170rneqd 5796 . . . . . . 7 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → ran (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))) = ran ( I ↾ 𝑎))
72 rnresi 5932 . . . . . . 7 ran ( I ↾ 𝑎) = 𝑎
7371, 72syl6eq 2875 . . . . . 6 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → ran (𝑧𝑎 ↦ (((0g𝐺) + 𝑧) (0g𝐺))) = 𝑎)
7455, 73eqtrd 2859 . . . . 5 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → ((0g𝐺) 𝑎) = 𝑎)
75 ovex 7184 . . . . . . . . . 10 ((𝑐 + 𝑧) 𝑐) ∈ V
76 oveq2 7159 . . . . . . . . . . 11 (𝑤 = ((𝑐 + 𝑧) 𝑐) → (𝑏 + 𝑤) = (𝑏 + ((𝑐 + 𝑧) 𝑐)))
7776oveq1d 7166 . . . . . . . . . 10 (𝑤 = ((𝑐 + 𝑧) 𝑐) → ((𝑏 + 𝑤) 𝑏) = ((𝑏 + ((𝑐 + 𝑧) 𝑐)) 𝑏))
7875, 77abrexco 6997 . . . . . . . . 9 {𝑢 ∣ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧𝑎 𝑣 = ((𝑐 + 𝑧) 𝑐)}𝑢 = ((𝑏 + 𝑤) 𝑏)} = {𝑢 ∣ ∃𝑧𝑎 𝑢 = ((𝑏 + ((𝑐 + 𝑧) 𝑐)) 𝑏)}
79 simprr 772 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → 𝑐𝑋)
80 simplr 768 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → 𝑎 ∈ (𝑃 pSyl 𝐺))
81 simpr 488 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑐𝑦 = 𝑎) → 𝑦 = 𝑎)
82 simpl 486 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑐𝑦 = 𝑎) → 𝑥 = 𝑐)
8382oveq1d 7166 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑐𝑦 = 𝑎) → (𝑥 + 𝑧) = (𝑐 + 𝑧))
8483, 82oveq12d 7169 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑐𝑦 = 𝑎) → ((𝑥 + 𝑧) 𝑥) = ((𝑐 + 𝑧) 𝑐))
8581, 84mpteq12dv 5138 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑐𝑦 = 𝑎) → (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐)))
8685rneqd 5796 . . . . . . . . . . . . . 14 ((𝑥 = 𝑐𝑦 = 𝑎) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = ran (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐)))
8751mptex 6979 . . . . . . . . . . . . . . 15 (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐)) ∈ V
8887rnex 7614 . . . . . . . . . . . . . 14 ran (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐)) ∈ V
8986, 37, 88ovmpoa 7300 . . . . . . . . . . . . 13 ((𝑐𝑋𝑎 ∈ (𝑃 pSyl 𝐺)) → (𝑐 𝑎) = ran (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐)))
9079, 80, 89syl2anc 587 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → (𝑐 𝑎) = ran (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐)))
91 eqid 2824 . . . . . . . . . . . . 13 (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐)) = (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐))
9291rnmpt 5815 . . . . . . . . . . . 12 ran (𝑧𝑎 ↦ ((𝑐 + 𝑧) 𝑐)) = {𝑣 ∣ ∃𝑧𝑎 𝑣 = ((𝑐 + 𝑧) 𝑐)}
9390, 92syl6eq 2875 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → (𝑐 𝑎) = {𝑣 ∣ ∃𝑧𝑎 𝑣 = ((𝑐 + 𝑧) 𝑐)})
9493rexeqdv 3403 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → (∃𝑤 ∈ (𝑐 𝑎)𝑢 = ((𝑏 + 𝑤) 𝑏) ↔ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧𝑎 𝑣 = ((𝑐 + 𝑧) 𝑐)}𝑢 = ((𝑏 + 𝑤) 𝑏)))
9594abbidv 2888 . . . . . . . . 9 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → {𝑢 ∣ ∃𝑤 ∈ (𝑐 𝑎)𝑢 = ((𝑏 + 𝑤) 𝑏)} = {𝑢 ∣ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧𝑎 𝑣 = ((𝑐 + 𝑧) 𝑐)}𝑢 = ((𝑏 + 𝑤) 𝑏)})
9640adantr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → 𝐺 ∈ Grp)
9796adantr 484 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → 𝐺 ∈ Grp)
98 simprl 770 . . . . . . . . . . . . . . . . 17 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → 𝑏𝑋)
996, 13grpcl 18113 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝑏𝑋𝑐𝑋) → (𝑏 + 𝑐) ∈ 𝑋)
10096, 98, 79, 99syl3anc 1368 . . . . . . . . . . . . . . . 16 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → (𝑏 + 𝑐) ∈ 𝑋)
101100adantr 484 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → (𝑏 + 𝑐) ∈ 𝑋)
10261adantlr 714 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → 𝑧𝑋)
1036, 13grpcl 18113 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ (𝑏 + 𝑐) ∈ 𝑋𝑧𝑋) → ((𝑏 + 𝑐) + 𝑧) ∈ 𝑋)
10497, 101, 102, 103syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → ((𝑏 + 𝑐) + 𝑧) ∈ 𝑋)
10579adantr 484 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → 𝑐𝑋)
10698adantr 484 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → 𝑏𝑋)
1076, 13, 14grpsubsub4 18194 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (((𝑏 + 𝑐) + 𝑧) ∈ 𝑋𝑐𝑋𝑏𝑋)) → ((((𝑏 + 𝑐) + 𝑧) 𝑐) 𝑏) = (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐)))
10897, 104, 105, 106, 107syl13anc 1369 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → ((((𝑏 + 𝑐) + 𝑧) 𝑐) 𝑏) = (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐)))
1096, 13grpass 18114 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ (𝑏𝑋𝑐𝑋𝑧𝑋)) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧)))
11097, 106, 105, 102, 109syl13anc 1369 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧)))
111110oveq1d 7166 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → (((𝑏 + 𝑐) + 𝑧) 𝑐) = ((𝑏 + (𝑐 + 𝑧)) 𝑐))
1126, 13grpcl 18113 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝑐𝑋𝑧𝑋) → (𝑐 + 𝑧) ∈ 𝑋)
11397, 105, 102, 112syl3anc 1368 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → (𝑐 + 𝑧) ∈ 𝑋)
1146, 13, 14grpaddsubass 18191 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ (𝑏𝑋 ∧ (𝑐 + 𝑧) ∈ 𝑋𝑐𝑋)) → ((𝑏 + (𝑐 + 𝑧)) 𝑐) = (𝑏 + ((𝑐 + 𝑧) 𝑐)))
11597, 106, 113, 105, 114syl13anc 1369 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → ((𝑏 + (𝑐 + 𝑧)) 𝑐) = (𝑏 + ((𝑐 + 𝑧) 𝑐)))
116111, 115eqtrd 2859 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → (((𝑏 + 𝑐) + 𝑧) 𝑐) = (𝑏 + ((𝑐 + 𝑧) 𝑐)))
117116oveq1d 7166 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → ((((𝑏 + 𝑐) + 𝑧) 𝑐) 𝑏) = ((𝑏 + ((𝑐 + 𝑧) 𝑐)) 𝑏))
118108, 117eqtr3d 2861 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐)) = ((𝑏 + ((𝑐 + 𝑧) 𝑐)) 𝑏))
119118eqeq2d 2835 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) ∧ 𝑧𝑎) → (𝑢 = (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐)) ↔ 𝑢 = ((𝑏 + ((𝑐 + 𝑧) 𝑐)) 𝑏)))
120119rexbidva 3288 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → (∃𝑧𝑎 𝑢 = (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐)) ↔ ∃𝑧𝑎 𝑢 = ((𝑏 + ((𝑐 + 𝑧) 𝑐)) 𝑏)))
121120abbidv 2888 . . . . . . . . 9 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → {𝑢 ∣ ∃𝑧𝑎 𝑢 = (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))} = {𝑢 ∣ ∃𝑧𝑎 𝑢 = ((𝑏 + ((𝑐 + 𝑧) 𝑐)) 𝑏)})
12278, 95, 1213eqtr4a 2885 . . . . . . . 8 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → {𝑢 ∣ ∃𝑤 ∈ (𝑐 𝑎)𝑢 = ((𝑏 + 𝑤) 𝑏)} = {𝑢 ∣ ∃𝑧𝑎 𝑢 = (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))})
123 eqid 2824 . . . . . . . . 9 (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)) = (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏))
124123rnmpt 5815 . . . . . . . 8 ran (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)) = {𝑢 ∣ ∃𝑤 ∈ (𝑐 𝑎)𝑢 = ((𝑏 + 𝑤) 𝑏)}
125 eqid 2824 . . . . . . . . 9 (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))) = (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐)))
126125rnmpt 5815 . . . . . . . 8 ran (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))) = {𝑢 ∣ ∃𝑧𝑎 𝑢 = (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))}
127122, 124, 1263eqtr4g 2884 . . . . . . 7 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → ran (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)) = ran (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))))
12839ad2antrr 725 . . . . . . . . 9 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺))
129128, 79, 80fovrnd 7316 . . . . . . . 8 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → (𝑐 𝑎) ∈ (𝑃 pSyl 𝐺))
130 simpr 488 . . . . . . . . . . . 12 ((𝑥 = 𝑏𝑦 = (𝑐 𝑎)) → 𝑦 = (𝑐 𝑎))
131 simpl 486 . . . . . . . . . . . . . 14 ((𝑥 = 𝑏𝑦 = (𝑐 𝑎)) → 𝑥 = 𝑏)
132131oveq1d 7166 . . . . . . . . . . . . 13 ((𝑥 = 𝑏𝑦 = (𝑐 𝑎)) → (𝑥 + 𝑧) = (𝑏 + 𝑧))
133132, 131oveq12d 7169 . . . . . . . . . . . 12 ((𝑥 = 𝑏𝑦 = (𝑐 𝑎)) → ((𝑥 + 𝑧) 𝑥) = ((𝑏 + 𝑧) 𝑏))
134130, 133mpteq12dv 5138 . . . . . . . . . . 11 ((𝑥 = 𝑏𝑦 = (𝑐 𝑎)) → (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = (𝑧 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑧) 𝑏)))
135 oveq2 7159 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → (𝑏 + 𝑧) = (𝑏 + 𝑤))
136135oveq1d 7166 . . . . . . . . . . . 12 (𝑧 = 𝑤 → ((𝑏 + 𝑧) 𝑏) = ((𝑏 + 𝑤) 𝑏))
137136cbvmptv 5156 . . . . . . . . . . 11 (𝑧 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑧) 𝑏)) = (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏))
138134, 137syl6eq 2875 . . . . . . . . . 10 ((𝑥 = 𝑏𝑦 = (𝑐 𝑎)) → (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)))
139138rneqd 5796 . . . . . . . . 9 ((𝑥 = 𝑏𝑦 = (𝑐 𝑎)) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = ran (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)))
140 ovex 7184 . . . . . . . . . . 11 (𝑐 𝑎) ∈ V
141140mptex 6979 . . . . . . . . . 10 (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)) ∈ V
142141rnex 7614 . . . . . . . . 9 ran (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)) ∈ V
143139, 37, 142ovmpoa 7300 . . . . . . . 8 ((𝑏𝑋 ∧ (𝑐 𝑎) ∈ (𝑃 pSyl 𝐺)) → (𝑏 (𝑐 𝑎)) = ran (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)))
14498, 129, 143syl2anc 587 . . . . . . 7 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → (𝑏 (𝑐 𝑎)) = ran (𝑤 ∈ (𝑐 𝑎) ↦ ((𝑏 + 𝑤) 𝑏)))
145 simpr 488 . . . . . . . . . . 11 ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎)
146 simpl 486 . . . . . . . . . . . . 13 ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑥 = (𝑏 + 𝑐))
147146oveq1d 7166 . . . . . . . . . . . 12 ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((𝑏 + 𝑐) + 𝑧))
148147, 146oveq12d 7169 . . . . . . . . . . 11 ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → ((𝑥 + 𝑧) 𝑥) = (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐)))
149145, 148mpteq12dv 5138 . . . . . . . . . 10 ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))))
150149rneqd 5796 . . . . . . . . 9 ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)) = ran (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))))
15151mptex 6979 . . . . . . . . . 10 (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))) ∈ V
152151rnex 7614 . . . . . . . . 9 ran (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))) ∈ V
153150, 37, 152ovmpoa 7300 . . . . . . . 8 (((𝑏 + 𝑐) ∈ 𝑋𝑎 ∈ (𝑃 pSyl 𝐺)) → ((𝑏 + 𝑐) 𝑎) = ran (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))))
154100, 80, 153syl2anc 587 . . . . . . 7 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → ((𝑏 + 𝑐) 𝑎) = ran (𝑧𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) (𝑏 + 𝑐))))
155127, 144, 1543eqtr4rd 2870 . . . . . 6 (((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏𝑋𝑐𝑋)) → ((𝑏 + 𝑐) 𝑎) = (𝑏 (𝑐 𝑎)))
156155ralrimivva 3186 . . . . 5 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → ∀𝑏𝑋𝑐𝑋 ((𝑏 + 𝑐) 𝑎) = (𝑏 (𝑐 𝑎)))
15774, 156jca 515 . . . 4 ((𝜑𝑎 ∈ (𝑃 pSyl 𝐺)) → (((0g𝐺) 𝑎) = 𝑎 ∧ ∀𝑏𝑋𝑐𝑋 ((𝑏 + 𝑐) 𝑎) = (𝑏 (𝑐 𝑎))))
158157ralrimiva 3177 . . 3 (𝜑 → ∀𝑎 ∈ (𝑃 pSyl 𝐺)(((0g𝐺) 𝑎) = 𝑎 ∧ ∀𝑏𝑋𝑐𝑋 ((𝑏 + 𝑐) 𝑎) = (𝑏 (𝑐 𝑎))))
15939, 158jca 515 . 2 (𝜑 → ( :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺) ∧ ∀𝑎 ∈ (𝑃 pSyl 𝐺)(((0g𝐺) 𝑎) = 𝑎 ∧ ∀𝑏𝑋𝑐𝑋 ((𝑏 + 𝑐) 𝑎) = (𝑏 (𝑐 𝑎)))))
1606, 13, 41isga 18423 . 2 ( ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)) ↔ ((𝐺 ∈ Grp ∧ (𝑃 pSyl 𝐺) ∈ V) ∧ ( :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺) ∧ ∀𝑎 ∈ (𝑃 pSyl 𝐺)(((0g𝐺) 𝑎) = 𝑎 ∧ ∀𝑏𝑋𝑐𝑋 ((𝑏 + 𝑐) 𝑎) = (𝑏 (𝑐 𝑎))))))
1613, 159, 160sylanbrc 586 1 (𝜑 ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {cab 2802  ∀wral 3133  ∃wrex 3134  Vcvv 3480   ⊆ wss 3919   class class class wbr 5053   ↦ cmpt 5133   I cid 5447   × cxp 5541  ran crn 5544   ↾ cres 5545  ⟶wf 6341  ‘cfv 6345  (class class class)co 7151   ∈ cmpo 7153   ≈ cen 8504  Fincfn 8507  ↑cexp 13436  ♯chash 13697  ℙcprime 16015   pCnt cpc 16173  Basecbs 16485  +gcplusg 16567  0gc0g 16715  Grpcgrp 18105  -gcsg 18107  SubGrpcsubg 18275   GrpAct cga 18421   pSyl cslw 18657 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-inf2 9103  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  ax-pre-sup 10615 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-disj 5019  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-isom 6354  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-2o 8101  df-oadd 8104  df-omul 8105  df-er 8287  df-ec 8289  df-qs 8293  df-map 8406  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-sup 8905  df-inf 8906  df-oi 8973  df-dju 9329  df-card 9367  df-acn 9370  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11637  df-2 11699  df-3 11700  df-n0 11897  df-xnn0 11967  df-z 11981  df-uz 12243  df-q 12348  df-rp 12389  df-fz 12897  df-fzo 13040  df-fl 13168  df-mod 13244  df-seq 13376  df-exp 13437  df-fac 13641  df-bc 13670  df-hash 13698  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-dvds 15610  df-gcd 15844  df-prm 16016  df-pc 16174  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-submnd 17959  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-eqg 18280  df-ghm 18358  df-ga 18422  df-od 18658  df-pgp 18660  df-slw 18661 This theorem is referenced by:  sylow3lem3  18756  sylow3lem5  18758
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