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Theorem sylow2blem1 19638
Description: Lemma for sylow2b 19641. 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 1138 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → 𝐵𝐻)
2 sylow2b.r . . . . 5 = (𝐺 ~QG 𝐾)
32ovexi 7465 . . . 4 ∈ V
4 simp3 1139 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → 𝐶𝑋)
5 ecelqsg 8812 . . . 4 (( ∈ V ∧ 𝐶𝑋) → [𝐶] ∈ (𝑋 / ))
63, 4, 5sylancr 587 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] ∈ (𝑋 / ))
7 simpr 484 . . . . . 6 ((𝑥 = 𝐵𝑦 = [𝐶] ) → 𝑦 = [𝐶] )
8 simpl 482 . . . . . . 7 ((𝑥 = 𝐵𝑦 = [𝐶] ) → 𝑥 = 𝐵)
98oveq1d 7446 . . . . . 6 ((𝑥 = 𝐵𝑦 = [𝐶] ) → (𝑥 + 𝑧) = (𝐵 + 𝑧))
107, 9mpteq12dv 5233 . . . . 5 ((𝑥 = 𝐵𝑦 = [𝐶] ) → (𝑧𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
1110rneqd 5949 . . . 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 8749 . . . . . . 7 ( ∈ V → [𝐶] ∈ V)
143, 13ax-mp 5 . . . . . 6 [𝐶] ∈ V
1514mptex 7243 . . . . 5 (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ∈ V
1615rnex 7932 . . . 4 ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ∈ V
1711, 12, 16ovmpoa 7588 . . 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 19196 . . . . . . 7 (𝐾 ∈ (SubGrp‘𝐺) → Er 𝑋)
2320, 22syl 17 . . . . . 6 (𝜑 Er 𝑋)
2423ecss 8793 . . . . 5 (𝜑 → [(𝐵 + 𝐶)] 𝑋)
2519, 24ssfid 9301 . . . 4 (𝜑 → [(𝐵 + 𝐶)] ∈ Fin)
26253ad2ant1 1134 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → [(𝐵 + 𝐶)] ∈ Fin)
27 vex 3484 . . . . . . . 8 𝑧 ∈ V
28 elecg 8789 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝐶𝑋) → (𝑧 ∈ [𝐶] 𝐶 𝑧))
2927, 4, 28sylancr 587 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] 𝐶 𝑧))
3029biimpa 476 . . . . . 6 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝑧 ∈ [𝐶] ) → 𝐶 𝑧)
31 sylow2b.h . . . . . . . . . . . 12 (𝜑𝐻 ∈ (SubGrp‘𝐺))
32 subgrcl 19149 . . . . . . . . . . . 12 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3331, 32syl 17 . . . . . . . . . . 11 (𝜑𝐺 ∈ Grp)
34333ad2ant1 1134 . . . . . . . . . 10 ((𝜑𝐵𝐻𝐶𝑋) → 𝐺 ∈ Grp)
3521subgss 19145 . . . . . . . . . . . . 13 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻𝑋)
3631, 35syl 17 . . . . . . . . . . . 12 (𝜑𝐻𝑋)
37363ad2ant1 1134 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → 𝐻𝑋)
3837, 1sseldd 3984 . . . . . . . . . 10 ((𝜑𝐵𝐻𝐶𝑋) → 𝐵𝑋)
39 sylow2b.a . . . . . . . . . . 11 + = (+g𝐺)
4021, 39grpcl 18959 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐶𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
4134, 38, 4, 40syl3anc 1373 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
4241adantr 480 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝐶) ∈ 𝑋)
4334adantr 480 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝐺 ∈ Grp)
4438adantr 480 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝐵𝑋)
4521subgss 19145 . . . . . . . . . . . . . 14 (𝐾 ∈ (SubGrp‘𝐺) → 𝐾𝑋)
4620, 45syl 17 . . . . . . . . . . . . 13 (𝜑𝐾𝑋)
47 eqid 2737 . . . . . . . . . . . . . 14 (invg𝐺) = (invg𝐺)
4821, 47, 39, 2eqgval 19195 . . . . . . . . . . . . 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 1134 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → (𝐶 𝑧 ↔ (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
5150biimpa 476 . . . . . . . . . 10 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾))
5251simp2d 1144 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝑧𝑋)
5321, 39grpcl 18959 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝑧𝑋) → (𝐵 + 𝑧) ∈ 𝑋)
5443, 44, 52, 53syl3anc 1373 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝑧) ∈ 𝑋)
5521, 47grpinvcl 19005 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝐵 + 𝐶) ∈ 𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
5634, 41, 55syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
5756adantr 480 . . . . . . . . . . 11 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
5821, 39grpass 18960 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋𝐵𝑋𝑧𝑋)) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
5943, 57, 44, 52, 58syl13anc 1374 . . . . . . . . . 10 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
6021, 39, 47grpinvadd 19036 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6134, 38, 4, 60syl3anc 1373 . . . . . . . . . . . . . . 15 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6221, 47grpinvcl 19005 . . . . . . . . . . . . . . . . 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 19003 . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . 13 ((𝜑𝐵𝐻𝐶𝑋) → (((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵))
6921, 39, 64grpnpcan 19050 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐶) ∈ 𝑋𝐵𝑋) → ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵) = ((invg𝐺)‘𝐶))
7034, 63, 38, 69syl3anc 1373 . . . . . . . . . . . . 13 ((𝜑𝐵𝐻𝐶𝑋) → ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵) = ((invg𝐺)‘𝐶))
7168, 70eqtrd 2777 . . . . . . . . . . . 12 ((𝜑𝐵𝐻𝐶𝑋) → (((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((invg𝐺)‘𝐶))
7271oveq1d 7446 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘𝐶) + 𝑧))
7372adantr 480 . . . . . . . . . 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 1145 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)
7674, 75eqeltrd 2841 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)
7721, 47, 39, 2eqgval 19195 . . . . . . . . . . 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 1134 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
8079adantr 480 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
8142, 54, 76, 80mpbir3and 1343 . . . . . . 7 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝐶) (𝐵 + 𝑧))
82 ovex 7464 . . . . . . . 8 (𝐵 + 𝑧) ∈ V
83 ovex 7464 . . . . . . . 8 (𝐵 + 𝐶) ∈ V
8482, 83elec 8791 . . . . . . 7 ((𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ↔ (𝐵 + 𝐶) (𝐵 + 𝑧))
8581, 84sylibr 234 . . . . . 6 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] )
8630, 85syldan 591 . . . . 5 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝑧 ∈ [𝐶] ) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] )
8786fmpttd 7135 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] ⟶[(𝐵 + 𝐶)] )
8887frnd 6744 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] )
89 eqid 2737 . . . . . . . . . . 11 (𝑧𝑋 ↦ (𝐵 + 𝑧)) = (𝑧𝑋 ↦ (𝐵 + 𝑧))
9021, 39, 89grplmulf1o 19031 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐵𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋)
9134, 38, 90syl2anc 584 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋)
92 f1of1 6847 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋 → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1𝑋)
9391, 92syl 17 . . . . . . . 8 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1𝑋)
9423ecss 8793 . . . . . . . . 9 (𝜑 → [𝐶] 𝑋)
95943ad2ant1 1134 . . . . . . . 8 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] 𝑋)
96 f1ssres 6811 . . . . . . . 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 6055 . . . . . . . 8 ([𝐶] 𝑋 → ((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
99 f1eq1 6799 . . . . . . . 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 232 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋)
102 f1f1orn 6859 . . . . . 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 9013 . . . . 5 ((𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1-onto→ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) → [𝐶] ≈ ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
105 ensym 9043 . . . . 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 1134 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ∈ (SubGrp‘𝐺))
10821, 2eqgen 19199 . . . . . . 7 ((𝐾 ∈ (SubGrp‘𝐺) ∧ [𝐶] ∈ (𝑋 / )) → 𝐾 ≈ [𝐶] )
109107, 6, 108syl2anc 584 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ≈ [𝐶] )
110 ensym 9043 . . . . . 6 (𝐾 ≈ [𝐶] → [𝐶] 𝐾)
111109, 110syl 17 . . . . 5 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] 𝐾)
112 ecelqsg 8812 . . . . . . 7 (( ∈ V ∧ (𝐵 + 𝐶) ∈ 𝑋) → [(𝐵 + 𝐶)] ∈ (𝑋 / ))
1133, 41, 112sylancr 587 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → [(𝐵 + 𝐶)] ∈ (𝑋 / ))
11421, 2eqgen 19199 . . . . . 6 ((𝐾 ∈ (SubGrp‘𝐺) ∧ [(𝐵 + 𝐶)] ∈ (𝑋 / )) → 𝐾 ≈ [(𝐵 + 𝐶)] )
115107, 113, 114syl2anc 584 . . . . 5 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ≈ [(𝐵 + 𝐶)] )
116 entr 9046 . . . . 5 (([𝐶] 𝐾𝐾 ≈ [(𝐵 + 𝐶)] ) → [𝐶] ≈ [(𝐵 + 𝐶)] )
117111, 115, 116syl2anc 584 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] ≈ [(𝐵 + 𝐶)] )
118 entr 9046 . . . 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 9293 . . 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 1373 . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  ran crn 5686  cres 5687  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433   Er wer 8742  [cec 8743   / cqs 8744  cen 8982  Fincfn 8985  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  invgcminusg 18952  -gcsg 18953  SubGrpcsubg 19138   ~QG cqg 19140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-qs 8751  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-eqg 19143
This theorem is referenced by:  sylow2blem2  19639  sylow2blem3  19640
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