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Theorem sylow2blem1 19292
Description: Lemma for sylow2b 19295. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
Hypotheses
Ref Expression
sylow2b.x 𝑋 = (Base‘𝐺)
sylow2b.xf (𝜑𝑋 ∈ Fin)
sylow2b.h (𝜑𝐻 ∈ (SubGrp‘𝐺))
sylow2b.k (𝜑𝐾 ∈ (SubGrp‘𝐺))
sylow2b.a + = (+g𝐺)
sylow2b.r = (𝐺 ~QG 𝐾)
sylow2b.m · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))
Assertion
Ref Expression
sylow2blem1 ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 · [𝐶] ) = [(𝐵 + 𝐶)] )
Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝐾,𝑦,𝑧   𝑥, · ,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥, ,𝑦,𝑧   𝜑,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sylow2blem1
StepHypRef Expression
1 simp2 1136 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → 𝐵𝐻)
2 sylow2b.r . . . . 5 = (𝐺 ~QG 𝐾)
32ovexi 7347 . . . 4 ∈ V
4 simp3 1137 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → 𝐶𝑋)
5 ecelqsg 8607 . . . 4 (( ∈ V ∧ 𝐶𝑋) → [𝐶] ∈ (𝑋 / ))
63, 4, 5sylancr 587 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] ∈ (𝑋 / ))
7 simpr 485 . . . . . 6 ((𝑥 = 𝐵𝑦 = [𝐶] ) → 𝑦 = [𝐶] )
8 simpl 483 . . . . . . 7 ((𝑥 = 𝐵𝑦 = [𝐶] ) → 𝑥 = 𝐵)
98oveq1d 7328 . . . . . 6 ((𝑥 = 𝐵𝑦 = [𝐶] ) → (𝑥 + 𝑧) = (𝐵 + 𝑧))
107, 9mpteq12dv 5176 . . . . 5 ((𝑥 = 𝐵𝑦 = [𝐶] ) → (𝑧𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
1110rneqd 5864 . . . 4 ((𝑥 = 𝐵𝑦 = [𝐶] ) → ran (𝑧𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
12 sylow2b.m . . . 4 · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))
13 ecexg 8548 . . . . . . 7 ( ∈ V → [𝐶] ∈ V)
143, 13ax-mp 5 . . . . . 6 [𝐶] ∈ V
1514mptex 7136 . . . . 5 (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ∈ V
1615rnex 7802 . . . 4 ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ∈ V
1711, 12, 16ovmpoa 7466 . . 3 ((𝐵𝐻 ∧ [𝐶] ∈ (𝑋 / )) → (𝐵 · [𝐶] ) = ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
181, 6, 17syl2anc 584 . 2 ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 · [𝐶] ) = ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
19 sylow2b.xf . . . . 5 (𝜑𝑋 ∈ Fin)
20 sylow2b.k . . . . . . 7 (𝜑𝐾 ∈ (SubGrp‘𝐺))
21 sylow2b.x . . . . . . . 8 𝑋 = (Base‘𝐺)
2221, 2eqger 18873 . . . . . . 7 (𝐾 ∈ (SubGrp‘𝐺) → Er 𝑋)
2320, 22syl 17 . . . . . 6 (𝜑 Er 𝑋)
2423ecss 8590 . . . . 5 (𝜑 → [(𝐵 + 𝐶)] 𝑋)
2519, 24ssfid 9107 . . . 4 (𝜑 → [(𝐵 + 𝐶)] ∈ Fin)
26253ad2ant1 1132 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → [(𝐵 + 𝐶)] ∈ Fin)
27 vex 3445 . . . . . . . 8 𝑧 ∈ V
28 elecg 8587 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝐶𝑋) → (𝑧 ∈ [𝐶] 𝐶 𝑧))
2927, 4, 28sylancr 587 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] 𝐶 𝑧))
3029biimpa 477 . . . . . 6 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝑧 ∈ [𝐶] ) → 𝐶 𝑧)
31 sylow2b.h . . . . . . . . . . . 12 (𝜑𝐻 ∈ (SubGrp‘𝐺))
32 subgrcl 18827 . . . . . . . . . . . 12 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3331, 32syl 17 . . . . . . . . . . 11 (𝜑𝐺 ∈ Grp)
34333ad2ant1 1132 . . . . . . . . . 10 ((𝜑𝐵𝐻𝐶𝑋) → 𝐺 ∈ Grp)
3521subgss 18823 . . . . . . . . . . . . 13 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻𝑋)
3631, 35syl 17 . . . . . . . . . . . 12 (𝜑𝐻𝑋)
37363ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → 𝐻𝑋)
3837, 1sseldd 3931 . . . . . . . . . 10 ((𝜑𝐵𝐻𝐶𝑋) → 𝐵𝑋)
39 sylow2b.a . . . . . . . . . . 11 + = (+g𝐺)
4021, 39grpcl 18652 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐶𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
4134, 38, 4, 40syl3anc 1370 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
4241adantr 481 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝐶) ∈ 𝑋)
4334adantr 481 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝐺 ∈ Grp)
4438adantr 481 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝐵𝑋)
4521subgss 18823 . . . . . . . . . . . . . 14 (𝐾 ∈ (SubGrp‘𝐺) → 𝐾𝑋)
4620, 45syl 17 . . . . . . . . . . . . 13 (𝜑𝐾𝑋)
47 eqid 2737 . . . . . . . . . . . . . 14 (invg𝐺) = (invg𝐺)
4821, 47, 39, 2eqgval 18872 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝐾𝑋) → (𝐶 𝑧 ↔ (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
4933, 46, 48syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑧 ↔ (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
50493ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → (𝐶 𝑧 ↔ (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
5150biimpa 477 . . . . . . . . . 10 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾))
5251simp2d 1142 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝑧𝑋)
5321, 39grpcl 18652 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝑧𝑋) → (𝐵 + 𝑧) ∈ 𝑋)
5443, 44, 52, 53syl3anc 1370 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝑧) ∈ 𝑋)
5521, 47grpinvcl 18694 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝐵 + 𝐶) ∈ 𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
5634, 41, 55syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
5756adantr 481 . . . . . . . . . . 11 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
5821, 39grpass 18653 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋𝐵𝑋𝑧𝑋)) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
5943, 57, 44, 52, 58syl13anc 1371 . . . . . . . . . 10 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
6021, 39, 47grpinvadd 18720 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6134, 38, 4, 60syl3anc 1370 . . . . . . . . . . . . . . 15 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6221, 47grpinvcl 18694 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝐶𝑋) → ((invg𝐺)‘𝐶) ∈ 𝑋)
6334, 4, 62syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘𝐶) ∈ 𝑋)
64 eqid 2737 . . . . . . . . . . . . . . . . 17 (-g𝐺) = (-g𝐺)
6521, 39, 47, 64grpsubval 18692 . . . . . . . . . . . . . . . 16 ((((invg𝐺)‘𝐶) ∈ 𝑋𝐵𝑋) → (((invg𝐺)‘𝐶)(-g𝐺)𝐵) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6663, 38, 65syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝐵𝐻𝐶𝑋) → (((invg𝐺)‘𝐶)(-g𝐺)𝐵) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6761, 66eqtr4d 2780 . . . . . . . . . . . . . 14 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) = (((invg𝐺)‘𝐶)(-g𝐺)𝐵))
6867oveq1d 7328 . . . . . . . . . . . . 13 ((𝜑𝐵𝐻𝐶𝑋) → (((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵))
6921, 39, 64grpnpcan 18734 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐶) ∈ 𝑋𝐵𝑋) → ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵) = ((invg𝐺)‘𝐶))
7034, 63, 38, 69syl3anc 1370 . . . . . . . . . . . . 13 ((𝜑𝐵𝐻𝐶𝑋) → ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵) = ((invg𝐺)‘𝐶))
7168, 70eqtrd 2777 . . . . . . . . . . . 12 ((𝜑𝐵𝐻𝐶𝑋) → (((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((invg𝐺)‘𝐶))
7271oveq1d 7328 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘𝐶) + 𝑧))
7372adantr 481 . . . . . . . . . 10 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘𝐶) + 𝑧))
7459, 73eqtr3d 2779 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) = (((invg𝐺)‘𝐶) + 𝑧))
7551simp3d 1143 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)
7674, 75eqeltrd 2838 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)
7721, 47, 39, 2eqgval 18872 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝐾𝑋) → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
7833, 46, 77syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
79783ad2ant1 1132 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
8079adantr 481 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
8142, 54, 76, 80mpbir3and 1341 . . . . . . 7 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝐶) (𝐵 + 𝑧))
82 ovex 7346 . . . . . . . 8 (𝐵 + 𝑧) ∈ V
83 ovex 7346 . . . . . . . 8 (𝐵 + 𝐶) ∈ V
8482, 83elec 8588 . . . . . . 7 ((𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ↔ (𝐵 + 𝐶) (𝐵 + 𝑧))
8581, 84sylibr 233 . . . . . 6 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] )
8630, 85syldan 591 . . . . 5 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝑧 ∈ [𝐶] ) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] )
8786fmpttd 7026 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] ⟶[(𝐵 + 𝐶)] )
8887frnd 6643 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] )
89 eqid 2737 . . . . . . . . . . 11 (𝑧𝑋 ↦ (𝐵 + 𝑧)) = (𝑧𝑋 ↦ (𝐵 + 𝑧))
9021, 39, 89grplmulf1o 18716 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐵𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋)
9134, 38, 90syl2anc 584 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋)
92 f1of1 6750 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋 → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1𝑋)
9391, 92syl 17 . . . . . . . 8 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1𝑋)
9423ecss 8590 . . . . . . . . 9 (𝜑 → [𝐶] 𝑋)
95943ad2ant1 1132 . . . . . . . 8 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] 𝑋)
96 f1ssres 6713 . . . . . . . 8 (((𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1𝑋 ∧ [𝐶] 𝑋) → ((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ):[𝐶] 1-1𝑋)
9793, 95, 96syl2anc 584 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → ((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ):[𝐶] 1-1𝑋)
98 resmpt 5962 . . . . . . . 8 ([𝐶] 𝑋 → ((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
99 f1eq1 6700 . . . . . . . 8 (((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) → (((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ):[𝐶] 1-1𝑋 ↔ (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋))
10095, 98, 993syl 18 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → (((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ):[𝐶] 1-1𝑋 ↔ (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋))
10197, 100mpbid 231 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋)
102 f1f1orn 6762 . . . . . 6 ((𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋 → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1-onto→ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
103101, 102syl 17 . . . . 5 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1-onto→ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
10414f1oen 8809 . . . . 5 ((𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1-onto→ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) → [𝐶] ≈ ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
105 ensym 8839 . . . . 5 ([𝐶] ≈ ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [𝐶] )
106103, 104, 1053syl 18 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [𝐶] )
107203ad2ant1 1132 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ∈ (SubGrp‘𝐺))
10821, 2eqgen 18876 . . . . . . 7 ((𝐾 ∈ (SubGrp‘𝐺) ∧ [𝐶] ∈ (𝑋 / )) → 𝐾 ≈ [𝐶] )
109107, 6, 108syl2anc 584 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ≈ [𝐶] )
110 ensym 8839 . . . . . 6 (𝐾 ≈ [𝐶] → [𝐶] 𝐾)
111109, 110syl 17 . . . . 5 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] 𝐾)
112 ecelqsg 8607 . . . . . . 7 (( ∈ V ∧ (𝐵 + 𝐶) ∈ 𝑋) → [(𝐵 + 𝐶)] ∈ (𝑋 / ))
1133, 41, 112sylancr 587 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → [(𝐵 + 𝐶)] ∈ (𝑋 / ))
11421, 2eqgen 18876 . . . . . 6 ((𝐾 ∈ (SubGrp‘𝐺) ∧ [(𝐵 + 𝐶)] ∈ (𝑋 / )) → 𝐾 ≈ [(𝐵 + 𝐶)] )
115107, 113, 114syl2anc 584 . . . . 5 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ≈ [(𝐵 + 𝐶)] )
116 entr 8842 . . . . 5 (([𝐶] 𝐾𝐾 ≈ [(𝐵 + 𝐶)] ) → [𝐶] ≈ [(𝐵 + 𝐶)] )
117111, 115, 116syl2anc 584 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] ≈ [(𝐵 + 𝐶)] )
118 entr 8842 . . . 4 ((ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∧ [𝐶] ≈ [(𝐵 + 𝐶)] ) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] )
119106, 117, 118syl2anc 584 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] )
120 fisseneq 9097 . . 3 (([(𝐵 + 𝐶)] ∈ Fin ∧ ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] ∧ ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] ) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) = [(𝐵 + 𝐶)] )
12126, 88, 119, 120syl3anc 1370 . 2 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) = [(𝐵 + 𝐶)] )
12218, 121eqtrd 2777 1 ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 · [𝐶] ) = [(𝐵 + 𝐶)] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  Vcvv 3441  wss 3896   class class class wbr 5085  cmpt 5168  ran crn 5606  cres 5607  1-1wf1 6460  1-1-ontowf1o 6462  cfv 6463  (class class class)co 7313  cmpo 7315   Er wer 8541  [cec 8542   / cqs 8543  cen 8776  Fincfn 8779  Basecbs 16979  +gcplusg 17029  Grpcgrp 18644  invgcminusg 18645  -gcsg 18646  SubGrpcsubg 18816   ~QG cqg 18818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626  ax-cnex 10997  ax-resscn 10998  ax-1cn 10999  ax-icn 11000  ax-addcl 11001  ax-addrcl 11002  ax-mulcl 11003  ax-mulrcl 11004  ax-mulcom 11005  ax-addass 11006  ax-mulass 11007  ax-distr 11008  ax-i2m1 11009  ax-1ne0 11010  ax-1rid 11011  ax-rnegex 11012  ax-rrecex 11013  ax-cnre 11014  ax-pre-lttri 11015  ax-pre-lttrn 11016  ax-pre-ltadd 11017  ax-pre-mulgt0 11018
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-pred 6222  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-riota 7270  df-ov 7316  df-oprab 7317  df-mpo 7318  df-om 7756  df-1st 7874  df-2nd 7875  df-frecs 8142  df-wrecs 8173  df-recs 8247  df-rdg 8286  df-1o 8342  df-er 8544  df-ec 8546  df-qs 8550  df-en 8780  df-dom 8781  df-sdom 8782  df-fin 8783  df-pnf 11081  df-mnf 11082  df-xr 11083  df-ltxr 11084  df-le 11085  df-sub 11277  df-neg 11278  df-nn 12044  df-2 12106  df-sets 16932  df-slot 16950  df-ndx 16962  df-base 16980  df-ress 17009  df-plusg 17042  df-0g 17219  df-mgm 18393  df-sgrp 18442  df-mnd 18453  df-grp 18647  df-minusg 18648  df-sbg 18649  df-subg 18819  df-eqg 18821
This theorem is referenced by:  sylow2blem2  19293  sylow2blem3  19294
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