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Theorem sylow2blem1 17963
Description: Lemma for sylow2b 17966. 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 1060 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → 𝐵𝐻)
2 sylow2b.r . . . . 5 = (𝐺 ~QG 𝐾)
3 ovex 6638 . . . . 5 (𝐺 ~QG 𝐾) ∈ V
42, 3eqeltri 2694 . . . 4 ∈ V
5 simp3 1061 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → 𝐶𝑋)
6 ecelqsg 7754 . . . 4 (( ∈ V ∧ 𝐶𝑋) → [𝐶] ∈ (𝑋 / ))
74, 5, 6sylancr 694 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] ∈ (𝑋 / ))
8 simpr 477 . . . . . 6 ((𝑥 = 𝐵𝑦 = [𝐶] ) → 𝑦 = [𝐶] )
9 simpl 473 . . . . . . 7 ((𝑥 = 𝐵𝑦 = [𝐶] ) → 𝑥 = 𝐵)
109oveq1d 6625 . . . . . 6 ((𝑥 = 𝐵𝑦 = [𝐶] ) → (𝑥 + 𝑧) = (𝐵 + 𝑧))
118, 10mpteq12dv 4698 . . . . 5 ((𝑥 = 𝐵𝑦 = [𝐶] ) → (𝑧𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
1211rneqd 5318 . . . 4 ((𝑥 = 𝐵𝑦 = [𝐶] ) → ran (𝑧𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
13 sylow2b.m . . . 4 · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))
14 ecexg 7698 . . . . . . 7 ( ∈ V → [𝐶] ∈ V)
154, 14ax-mp 5 . . . . . 6 [𝐶] ∈ V
1615mptex 6446 . . . . 5 (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ∈ V
1716rnex 7054 . . . 4 ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ∈ V
1812, 13, 17ovmpt2a 6751 . . 3 ((𝐵𝐻 ∧ [𝐶] ∈ (𝑋 / )) → (𝐵 · [𝐶] ) = ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
191, 7, 18syl2anc 692 . 2 ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 · [𝐶] ) = ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
20 sylow2b.xf . . . . 5 (𝜑𝑋 ∈ Fin)
21 sylow2b.k . . . . . . 7 (𝜑𝐾 ∈ (SubGrp‘𝐺))
22 sylow2b.x . . . . . . . 8 𝑋 = (Base‘𝐺)
2322, 2eqger 17572 . . . . . . 7 (𝐾 ∈ (SubGrp‘𝐺) → Er 𝑋)
2421, 23syl 17 . . . . . 6 (𝜑 Er 𝑋)
2524ecss 7740 . . . . 5 (𝜑 → [(𝐵 + 𝐶)] 𝑋)
26 ssfi 8131 . . . . 5 ((𝑋 ∈ Fin ∧ [(𝐵 + 𝐶)] 𝑋) → [(𝐵 + 𝐶)] ∈ Fin)
2720, 25, 26syl2anc 692 . . . 4 (𝜑 → [(𝐵 + 𝐶)] ∈ Fin)
28273ad2ant1 1080 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → [(𝐵 + 𝐶)] ∈ Fin)
29 vex 3192 . . . . . . . 8 𝑧 ∈ V
30 elecg 7737 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝐶𝑋) → (𝑧 ∈ [𝐶] 𝐶 𝑧))
3129, 5, 30sylancr 694 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] 𝐶 𝑧))
3231biimpa 501 . . . . . 6 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝑧 ∈ [𝐶] ) → 𝐶 𝑧)
33 sylow2b.h . . . . . . . . . . . 12 (𝜑𝐻 ∈ (SubGrp‘𝐺))
34 subgrcl 17527 . . . . . . . . . . . 12 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3533, 34syl 17 . . . . . . . . . . 11 (𝜑𝐺 ∈ Grp)
36353ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝐵𝐻𝐶𝑋) → 𝐺 ∈ Grp)
3722subgss 17523 . . . . . . . . . . . . 13 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻𝑋)
3833, 37syl 17 . . . . . . . . . . . 12 (𝜑𝐻𝑋)
39383ad2ant1 1080 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → 𝐻𝑋)
4039, 1sseldd 3588 . . . . . . . . . 10 ((𝜑𝐵𝐻𝐶𝑋) → 𝐵𝑋)
41 sylow2b.a . . . . . . . . . . 11 + = (+g𝐺)
4222, 41grpcl 17358 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐶𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
4336, 40, 5, 42syl3anc 1323 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
4443adantr 481 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝐶) ∈ 𝑋)
4536adantr 481 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝐺 ∈ Grp)
4640adantr 481 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝐵𝑋)
4722subgss 17523 . . . . . . . . . . . . . 14 (𝐾 ∈ (SubGrp‘𝐺) → 𝐾𝑋)
4821, 47syl 17 . . . . . . . . . . . . 13 (𝜑𝐾𝑋)
49 eqid 2621 . . . . . . . . . . . . . 14 (invg𝐺) = (invg𝐺)
5022, 49, 41, 2eqgval 17571 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝐾𝑋) → (𝐶 𝑧 ↔ (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
5135, 48, 50syl2anc 692 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑧 ↔ (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
52513ad2ant1 1080 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → (𝐶 𝑧 ↔ (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
5352biimpa 501 . . . . . . . . . 10 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐶𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾))
5453simp2d 1072 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → 𝑧𝑋)
5522, 41grpcl 17358 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝑧𝑋) → (𝐵 + 𝑧) ∈ 𝑋)
5645, 46, 54, 55syl3anc 1323 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝑧) ∈ 𝑋)
5722, 49grpinvcl 17395 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝐵 + 𝐶) ∈ 𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
5836, 43, 57syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
5958adantr 481 . . . . . . . . . . 11 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
6022, 41grpass 17359 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋𝐵𝑋𝑧𝑋)) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
6145, 59, 46, 54, 60syl13anc 1325 . . . . . . . . . 10 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
6222, 41, 49grpinvadd 17421 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6336, 40, 5, 62syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6422, 49grpinvcl 17395 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝐶𝑋) → ((invg𝐺)‘𝐶) ∈ 𝑋)
6536, 5, 64syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘𝐶) ∈ 𝑋)
66 eqid 2621 . . . . . . . . . . . . . . . . 17 (-g𝐺) = (-g𝐺)
6722, 41, 49, 66grpsubval 17393 . . . . . . . . . . . . . . . 16 ((((invg𝐺)‘𝐶) ∈ 𝑋𝐵𝑋) → (((invg𝐺)‘𝐶)(-g𝐺)𝐵) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6865, 40, 67syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝐵𝐻𝐶𝑋) → (((invg𝐺)‘𝐶)(-g𝐺)𝐵) = (((invg𝐺)‘𝐶) + ((invg𝐺)‘𝐵)))
6963, 68eqtr4d 2658 . . . . . . . . . . . . . 14 ((𝜑𝐵𝐻𝐶𝑋) → ((invg𝐺)‘(𝐵 + 𝐶)) = (((invg𝐺)‘𝐶)(-g𝐺)𝐵))
7069oveq1d 6625 . . . . . . . . . . . . 13 ((𝜑𝐵𝐻𝐶𝑋) → (((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵))
7122, 41, 66grpnpcan 17435 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐶) ∈ 𝑋𝐵𝑋) → ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵) = ((invg𝐺)‘𝐶))
7236, 65, 40, 71syl3anc 1323 . . . . . . . . . . . . 13 ((𝜑𝐵𝐻𝐶𝑋) → ((((invg𝐺)‘𝐶)(-g𝐺)𝐵) + 𝐵) = ((invg𝐺)‘𝐶))
7370, 72eqtrd 2655 . . . . . . . . . . . 12 ((𝜑𝐵𝐻𝐶𝑋) → (((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((invg𝐺)‘𝐶))
7473oveq1d 6625 . . . . . . . . . . 11 ((𝜑𝐵𝐻𝐶𝑋) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘𝐶) + 𝑧))
7574adantr 481 . . . . . . . . . 10 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((((invg𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg𝐺)‘𝐶) + 𝑧))
7661, 75eqtr3d 2657 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) = (((invg𝐺)‘𝐶) + 𝑧))
7753simp3d 1073 . . . . . . . . 9 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (((invg𝐺)‘𝐶) + 𝑧) ∈ 𝐾)
7876, 77eqeltrd 2698 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)
7922, 49, 41, 2eqgval 17571 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝐾𝑋) → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
8035, 48, 79syl2anc 692 . . . . . . . . . 10 (𝜑 → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
81803ad2ant1 1080 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
8281adantr 481 . . . . . . . 8 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → ((𝐵 + 𝐶) (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
8344, 56, 78, 82mpbir3and 1243 . . . . . . 7 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝐶) (𝐵 + 𝑧))
84 ovex 6638 . . . . . . . 8 (𝐵 + 𝑧) ∈ V
85 ovex 6638 . . . . . . . 8 (𝐵 + 𝐶) ∈ V
8684, 85elec 7738 . . . . . . 7 ((𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ↔ (𝐵 + 𝐶) (𝐵 + 𝑧))
8783, 86sylibr 224 . . . . . 6 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝐶 𝑧) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] )
8832, 87syldan 487 . . . . 5 (((𝜑𝐵𝐻𝐶𝑋) ∧ 𝑧 ∈ [𝐶] ) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] )
89 eqid 2621 . . . . 5 (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧))
9088, 89fmptd 6346 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] ⟶[(𝐵 + 𝐶)] )
91 frn 6015 . . . 4 ((𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] ⟶[(𝐵 + 𝐶)] → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] )
9290, 91syl 17 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] )
93 eqid 2621 . . . . . . . . . . 11 (𝑧𝑋 ↦ (𝐵 + 𝑧)) = (𝑧𝑋 ↦ (𝐵 + 𝑧))
9422, 41, 93grplmulf1o 17417 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐵𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋)
9536, 40, 94syl2anc 692 . . . . . . . . 9 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋)
96 f1of1 6098 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1-onto𝑋 → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1𝑋)
9795, 96syl 17 . . . . . . . 8 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1𝑋)
9824ecss 7740 . . . . . . . . 9 (𝜑 → [𝐶] 𝑋)
99983ad2ant1 1080 . . . . . . . 8 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] 𝑋)
100 f1ssres 6070 . . . . . . . 8 (((𝑧𝑋 ↦ (𝐵 + 𝑧)):𝑋1-1𝑋 ∧ [𝐶] 𝑋) → ((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ):[𝐶] 1-1𝑋)
10197, 99, 100syl2anc 692 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → ((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ):[𝐶] 1-1𝑋)
102 resmpt 5413 . . . . . . . 8 ([𝐶] 𝑋 → ((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
103 f1eq1 6058 . . . . . . . 8 (((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ) = (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) → (((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ):[𝐶] 1-1𝑋 ↔ (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋))
10499, 102, 1033syl 18 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → (((𝑧𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ):[𝐶] 1-1𝑋 ↔ (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋))
105101, 104mpbid 222 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋)
106 f1f1orn 6110 . . . . . 6 ((𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1𝑋 → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1-onto→ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
107105, 106syl 17 . . . . 5 ((𝜑𝐵𝐻𝐶𝑋) → (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1-onto→ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
10815f1oen 7927 . . . . 5 ((𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)):[𝐶] 1-1-onto→ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) → [𝐶] ≈ ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)))
109 ensym 7956 . . . . 5 ([𝐶] ≈ ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [𝐶] )
110107, 108, 1093syl 18 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [𝐶] )
111213ad2ant1 1080 . . . . . . 7 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ∈ (SubGrp‘𝐺))
11222, 2eqgen 17575 . . . . . . 7 ((𝐾 ∈ (SubGrp‘𝐺) ∧ [𝐶] ∈ (𝑋 / )) → 𝐾 ≈ [𝐶] )
113111, 7, 112syl2anc 692 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ≈ [𝐶] )
114 ensym 7956 . . . . . 6 (𝐾 ≈ [𝐶] → [𝐶] 𝐾)
115113, 114syl 17 . . . . 5 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] 𝐾)
116 ecelqsg 7754 . . . . . . 7 (( ∈ V ∧ (𝐵 + 𝐶) ∈ 𝑋) → [(𝐵 + 𝐶)] ∈ (𝑋 / ))
1174, 43, 116sylancr 694 . . . . . 6 ((𝜑𝐵𝐻𝐶𝑋) → [(𝐵 + 𝐶)] ∈ (𝑋 / ))
11822, 2eqgen 17575 . . . . . 6 ((𝐾 ∈ (SubGrp‘𝐺) ∧ [(𝐵 + 𝐶)] ∈ (𝑋 / )) → 𝐾 ≈ [(𝐵 + 𝐶)] )
119111, 117, 118syl2anc 692 . . . . 5 ((𝜑𝐵𝐻𝐶𝑋) → 𝐾 ≈ [(𝐵 + 𝐶)] )
120 entr 7959 . . . . 5 (([𝐶] 𝐾𝐾 ≈ [(𝐵 + 𝐶)] ) → [𝐶] ≈ [(𝐵 + 𝐶)] )
121115, 119, 120syl2anc 692 . . . 4 ((𝜑𝐵𝐻𝐶𝑋) → [𝐶] ≈ [(𝐵 + 𝐶)] )
122 entr 7959 . . . 4 ((ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∧ [𝐶] ≈ [(𝐵 + 𝐶)] ) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] )
123110, 121, 122syl2anc 692 . . 3 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] )
124 fisseneq 8122 . . 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)] )
12528, 92, 123, 124syl3anc 1323 . 2 ((𝜑𝐵𝐻𝐶𝑋) → ran (𝑧 ∈ [𝐶] ↦ (𝐵 + 𝑧)) = [(𝐵 + 𝐶)] )
12619, 125eqtrd 2655 1 ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 · [𝐶] ) = [(𝐵 + 𝐶)] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  wss 3559   class class class wbr 4618  cmpt 4678  ran crn 5080  cres 5081  wf 5848  1-1wf1 5849  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cmpt2 6612   Er wer 7691  [cec 7692   / cqs 7693  cen 7903  Fincfn 7906  Basecbs 15788  +gcplusg 15869  Grpcgrp 17350  invgcminusg 17351  -gcsg 17352  SubGrpcsubg 17516   ~QG cqg 17518
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-ec 7696  df-qs 7700  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-0g 16030  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-grp 17353  df-minusg 17354  df-sbg 17355  df-subg 17519  df-eqg 17521
This theorem is referenced by:  sylow2blem2  17964  sylow2blem3  17965
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