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Theorem sylow2blem1 19490
Description: Lemma for sylow2b 19493. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
Hypotheses
Ref Expression
sylow2b.x 𝑋 = (Baseβ€˜πΊ)
sylow2b.xf (πœ‘ β†’ 𝑋 ∈ Fin)
sylow2b.h (πœ‘ β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
sylow2b.k (πœ‘ β†’ 𝐾 ∈ (SubGrpβ€˜πΊ))
sylow2b.a + = (+gβ€˜πΊ)
sylow2b.r ∼ = (𝐺 ~QG 𝐾)
sylow2b.m Β· = (π‘₯ ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (π‘₯ + 𝑧)))
Assertion
Ref Expression
sylow2blem1 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡 Β· [𝐢] ∼ ) = [(𝐡 + 𝐢)] ∼ )
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐺   π‘₯,𝐾,𝑦,𝑧   π‘₯, Β· ,𝑦,𝑧   π‘₯, + ,𝑦,𝑧   π‘₯, ∼ ,𝑦,𝑧   πœ‘,𝑧   π‘₯,𝐡,𝑦,𝑧   π‘₯,𝐢,𝑦,𝑧   π‘₯,𝐻,𝑦,𝑧   π‘₯,𝑋,𝑦,𝑧
Allowed substitution hints:   πœ‘(π‘₯,𝑦)

Proof of Theorem sylow2blem1
StepHypRef Expression
1 simp2 1137 . . 3 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ 𝐡 ∈ 𝐻)
2 sylow2b.r . . . . 5 ∼ = (𝐺 ~QG 𝐾)
32ovexi 7445 . . . 4 ∼ ∈ V
4 simp3 1138 . . . 4 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ 𝐢 ∈ 𝑋)
5 ecelqsg 8768 . . . 4 (( ∼ ∈ V ∧ 𝐢 ∈ 𝑋) β†’ [𝐢] ∼ ∈ (𝑋 / ∼ ))
63, 4, 5sylancr 587 . . 3 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ [𝐢] ∼ ∈ (𝑋 / ∼ ))
7 simpr 485 . . . . . 6 ((π‘₯ = 𝐡 ∧ 𝑦 = [𝐢] ∼ ) β†’ 𝑦 = [𝐢] ∼ )
8 simpl 483 . . . . . . 7 ((π‘₯ = 𝐡 ∧ 𝑦 = [𝐢] ∼ ) β†’ π‘₯ = 𝐡)
98oveq1d 7426 . . . . . 6 ((π‘₯ = 𝐡 ∧ 𝑦 = [𝐢] ∼ ) β†’ (π‘₯ + 𝑧) = (𝐡 + 𝑧))
107, 9mpteq12dv 5239 . . . . 5 ((π‘₯ = 𝐡 ∧ 𝑦 = [𝐢] ∼ ) β†’ (𝑧 ∈ 𝑦 ↦ (π‘₯ + 𝑧)) = (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)))
1110rneqd 5937 . . . 4 ((π‘₯ = 𝐡 ∧ 𝑦 = [𝐢] ∼ ) β†’ ran (𝑧 ∈ 𝑦 ↦ (π‘₯ + 𝑧)) = ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)))
12 sylow2b.m . . . 4 Β· = (π‘₯ ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (π‘₯ + 𝑧)))
13 ecexg 8709 . . . . . . 7 ( ∼ ∈ V β†’ [𝐢] ∼ ∈ V)
143, 13ax-mp 5 . . . . . 6 [𝐢] ∼ ∈ V
1514mptex 7227 . . . . 5 (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) ∈ V
1615rnex 7905 . . . 4 ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) ∈ V
1711, 12, 16ovmpoa 7565 . . 3 ((𝐡 ∈ 𝐻 ∧ [𝐢] ∼ ∈ (𝑋 / ∼ )) β†’ (𝐡 Β· [𝐢] ∼ ) = ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)))
181, 6, 17syl2anc 584 . 2 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡 Β· [𝐢] ∼ ) = ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)))
19 sylow2b.xf . . . . 5 (πœ‘ β†’ 𝑋 ∈ Fin)
20 sylow2b.k . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ (SubGrpβ€˜πΊ))
21 sylow2b.x . . . . . . . 8 𝑋 = (Baseβ€˜πΊ)
2221, 2eqger 19060 . . . . . . 7 (𝐾 ∈ (SubGrpβ€˜πΊ) β†’ ∼ Er 𝑋)
2320, 22syl 17 . . . . . 6 (πœ‘ β†’ ∼ Er 𝑋)
2423ecss 8751 . . . . 5 (πœ‘ β†’ [(𝐡 + 𝐢)] ∼ βŠ† 𝑋)
2519, 24ssfid 9269 . . . 4 (πœ‘ β†’ [(𝐡 + 𝐢)] ∼ ∈ Fin)
26253ad2ant1 1133 . . 3 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ [(𝐡 + 𝐢)] ∼ ∈ Fin)
27 vex 3478 . . . . . . . 8 𝑧 ∈ V
28 elecg 8748 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝐢 ∈ 𝑋) β†’ (𝑧 ∈ [𝐢] ∼ ↔ 𝐢 ∼ 𝑧))
2927, 4, 28sylancr 587 . . . . . . 7 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝑧 ∈ [𝐢] ∼ ↔ 𝐢 ∼ 𝑧))
3029biimpa 477 . . . . . 6 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝑧 ∈ [𝐢] ∼ ) β†’ 𝐢 ∼ 𝑧)
31 sylow2b.h . . . . . . . . . . . 12 (πœ‘ β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
32 subgrcl 19013 . . . . . . . . . . . 12 (𝐻 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
3331, 32syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 ∈ Grp)
34333ad2ant1 1133 . . . . . . . . . 10 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ 𝐺 ∈ Grp)
3521subgss 19009 . . . . . . . . . . . . 13 (𝐻 ∈ (SubGrpβ€˜πΊ) β†’ 𝐻 βŠ† 𝑋)
3631, 35syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐻 βŠ† 𝑋)
37363ad2ant1 1133 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ 𝐻 βŠ† 𝑋)
3837, 1sseldd 3983 . . . . . . . . . 10 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ 𝐡 ∈ 𝑋)
39 sylow2b.a . . . . . . . . . . 11 + = (+gβ€˜πΊ)
4021, 39grpcl 18829 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡 + 𝐢) ∈ 𝑋)
4134, 38, 4, 40syl3anc 1371 . . . . . . . . 9 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡 + 𝐢) ∈ 𝑋)
4241adantr 481 . . . . . . . 8 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ (𝐡 + 𝐢) ∈ 𝑋)
4334adantr 481 . . . . . . . . 9 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ 𝐺 ∈ Grp)
4438adantr 481 . . . . . . . . 9 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ 𝐡 ∈ 𝑋)
4521subgss 19009 . . . . . . . . . . . . . 14 (𝐾 ∈ (SubGrpβ€˜πΊ) β†’ 𝐾 βŠ† 𝑋)
4620, 45syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐾 βŠ† 𝑋)
47 eqid 2732 . . . . . . . . . . . . . 14 (invgβ€˜πΊ) = (invgβ€˜πΊ)
4821, 47, 39, 2eqgval 19059 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝐾 βŠ† 𝑋) β†’ (𝐢 ∼ 𝑧 ↔ (𝐢 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜πΆ) + 𝑧) ∈ 𝐾)))
4933, 46, 48syl2anc 584 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐢 ∼ 𝑧 ↔ (𝐢 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜πΆ) + 𝑧) ∈ 𝐾)))
50493ad2ant1 1133 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝐢 ∼ 𝑧 ↔ (𝐢 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜πΆ) + 𝑧) ∈ 𝐾)))
5150biimpa 477 . . . . . . . . . 10 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ (𝐢 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜πΆ) + 𝑧) ∈ 𝐾))
5251simp2d 1143 . . . . . . . . 9 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ 𝑧 ∈ 𝑋)
5321, 39grpcl 18829 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐡 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) β†’ (𝐡 + 𝑧) ∈ 𝑋)
5443, 44, 52, 53syl3anc 1371 . . . . . . . 8 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ (𝐡 + 𝑧) ∈ 𝑋)
5521, 47grpinvcl 18874 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝐡 + 𝐢) ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) ∈ 𝑋)
5634, 41, 55syl2anc 584 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) ∈ 𝑋)
5756adantr 481 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ ((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) ∈ 𝑋)
5821, 39grpass 18830 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + 𝐡) + 𝑧) = (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + (𝐡 + 𝑧)))
5943, 57, 44, 52, 58syl13anc 1372 . . . . . . . . . 10 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ ((((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + 𝐡) + 𝑧) = (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + (𝐡 + 𝑧)))
6021, 39, 47grpinvadd 18903 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) = (((invgβ€˜πΊ)β€˜πΆ) + ((invgβ€˜πΊ)β€˜π΅)))
6134, 38, 4, 60syl3anc 1371 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) = (((invgβ€˜πΊ)β€˜πΆ) + ((invgβ€˜πΊ)β€˜π΅)))
6221, 47grpinvcl 18874 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝐢 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
6334, 4, 62syl2anc 584 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
64 eqid 2732 . . . . . . . . . . . . . . . . 17 (-gβ€˜πΊ) = (-gβ€˜πΊ)
6521, 39, 47, 64grpsubval 18872 . . . . . . . . . . . . . . . 16 ((((invgβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜πΆ)(-gβ€˜πΊ)𝐡) = (((invgβ€˜πΊ)β€˜πΆ) + ((invgβ€˜πΊ)β€˜π΅)))
6663, 38, 65syl2anc 584 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜πΆ)(-gβ€˜πΊ)𝐡) = (((invgβ€˜πΊ)β€˜πΆ) + ((invgβ€˜πΊ)β€˜π΅)))
6761, 66eqtr4d 2775 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) = (((invgβ€˜πΊ)β€˜πΆ)(-gβ€˜πΊ)𝐡))
6867oveq1d 7426 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + 𝐡) = ((((invgβ€˜πΊ)β€˜πΆ)(-gβ€˜πΊ)𝐡) + 𝐡))
6921, 39, 64grpnpcan 18917 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ ((invgβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((((invgβ€˜πΊ)β€˜πΆ)(-gβ€˜πΊ)𝐡) + 𝐡) = ((invgβ€˜πΊ)β€˜πΆ))
7034, 63, 38, 69syl3anc 1371 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ((((invgβ€˜πΊ)β€˜πΆ)(-gβ€˜πΊ)𝐡) + 𝐡) = ((invgβ€˜πΊ)β€˜πΆ))
7168, 70eqtrd 2772 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + 𝐡) = ((invgβ€˜πΊ)β€˜πΆ))
7271oveq1d 7426 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ((((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + 𝐡) + 𝑧) = (((invgβ€˜πΊ)β€˜πΆ) + 𝑧))
7372adantr 481 . . . . . . . . . 10 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ ((((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + 𝐡) + 𝑧) = (((invgβ€˜πΊ)β€˜πΆ) + 𝑧))
7459, 73eqtr3d 2774 . . . . . . . . 9 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + (𝐡 + 𝑧)) = (((invgβ€˜πΊ)β€˜πΆ) + 𝑧))
7551simp3d 1144 . . . . . . . . 9 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ (((invgβ€˜πΊ)β€˜πΆ) + 𝑧) ∈ 𝐾)
7674, 75eqeltrd 2833 . . . . . . . 8 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + (𝐡 + 𝑧)) ∈ 𝐾)
7721, 47, 39, 2eqgval 19059 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝐾 βŠ† 𝑋) β†’ ((𝐡 + 𝐢) ∼ (𝐡 + 𝑧) ↔ ((𝐡 + 𝐢) ∈ 𝑋 ∧ (𝐡 + 𝑧) ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + (𝐡 + 𝑧)) ∈ 𝐾)))
7833, 46, 77syl2anc 584 . . . . . . . . . 10 (πœ‘ β†’ ((𝐡 + 𝐢) ∼ (𝐡 + 𝑧) ↔ ((𝐡 + 𝐢) ∈ 𝑋 ∧ (𝐡 + 𝑧) ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + (𝐡 + 𝑧)) ∈ 𝐾)))
79783ad2ant1 1133 . . . . . . . . 9 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ((𝐡 + 𝐢) ∼ (𝐡 + 𝑧) ↔ ((𝐡 + 𝐢) ∈ 𝑋 ∧ (𝐡 + 𝑧) ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + (𝐡 + 𝑧)) ∈ 𝐾)))
8079adantr 481 . . . . . . . 8 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ ((𝐡 + 𝐢) ∼ (𝐡 + 𝑧) ↔ ((𝐡 + 𝐢) ∈ 𝑋 ∧ (𝐡 + 𝑧) ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜(𝐡 + 𝐢)) + (𝐡 + 𝑧)) ∈ 𝐾)))
8142, 54, 76, 80mpbir3and 1342 . . . . . . 7 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ (𝐡 + 𝐢) ∼ (𝐡 + 𝑧))
82 ovex 7444 . . . . . . . 8 (𝐡 + 𝑧) ∈ V
83 ovex 7444 . . . . . . . 8 (𝐡 + 𝐢) ∈ V
8482, 83elec 8749 . . . . . . 7 ((𝐡 + 𝑧) ∈ [(𝐡 + 𝐢)] ∼ ↔ (𝐡 + 𝐢) ∼ (𝐡 + 𝑧))
8581, 84sylibr 233 . . . . . 6 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝐢 ∼ 𝑧) β†’ (𝐡 + 𝑧) ∈ [(𝐡 + 𝐢)] ∼ )
8630, 85syldan 591 . . . . 5 (((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) ∧ 𝑧 ∈ [𝐢] ∼ ) β†’ (𝐡 + 𝑧) ∈ [(𝐡 + 𝐢)] ∼ )
8786fmpttd 7116 . . . 4 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)):[𝐢] ∼ ⟢[(𝐡 + 𝐢)] ∼ )
8887frnd 6725 . . 3 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) βŠ† [(𝐡 + 𝐢)] ∼ )
89 eqid 2732 . . . . . . . . . . 11 (𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧))
9021, 39, 89grplmulf1o 18899 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐡 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)):𝑋–1-1-onto→𝑋)
9134, 38, 90syl2anc 584 . . . . . . . . 9 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)):𝑋–1-1-onto→𝑋)
92 f1of1 6832 . . . . . . . . 9 ((𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)):𝑋–1-1-onto→𝑋 β†’ (𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)):𝑋–1-1→𝑋)
9391, 92syl 17 . . . . . . . 8 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)):𝑋–1-1→𝑋)
9423ecss 8751 . . . . . . . . 9 (πœ‘ β†’ [𝐢] ∼ βŠ† 𝑋)
95943ad2ant1 1133 . . . . . . . 8 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ [𝐢] ∼ βŠ† 𝑋)
96 f1ssres 6795 . . . . . . . 8 (((𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)):𝑋–1-1→𝑋 ∧ [𝐢] ∼ βŠ† 𝑋) β†’ ((𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)) β†Ύ [𝐢] ∼ ):[𝐢] ∼ –1-1→𝑋)
9793, 95, 96syl2anc 584 . . . . . . 7 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ((𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)) β†Ύ [𝐢] ∼ ):[𝐢] ∼ –1-1→𝑋)
98 resmpt 6037 . . . . . . . 8 ([𝐢] ∼ βŠ† 𝑋 β†’ ((𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)) β†Ύ [𝐢] ∼ ) = (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)))
99 f1eq1 6782 . . . . . . . 8 (((𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)) β†Ύ [𝐢] ∼ ) = (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β†’ (((𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)) β†Ύ [𝐢] ∼ ):[𝐢] ∼ –1-1→𝑋 ↔ (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)):[𝐢] ∼ –1-1→𝑋))
10095, 98, 993syl 18 . . . . . . 7 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (((𝑧 ∈ 𝑋 ↦ (𝐡 + 𝑧)) β†Ύ [𝐢] ∼ ):[𝐢] ∼ –1-1→𝑋 ↔ (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)):[𝐢] ∼ –1-1→𝑋))
10197, 100mpbid 231 . . . . . 6 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)):[𝐢] ∼ –1-1→𝑋)
102 f1f1orn 6844 . . . . . 6 ((𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)):[𝐢] ∼ –1-1→𝑋 β†’ (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)):[𝐢] ∼ –1-1-ontoβ†’ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)))
103101, 102syl 17 . . . . 5 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)):[𝐢] ∼ –1-1-ontoβ†’ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)))
10414f1oen 8971 . . . . 5 ((𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)):[𝐢] ∼ –1-1-ontoβ†’ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β†’ [𝐢] ∼ β‰ˆ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)))
105 ensym 9001 . . . . 5 ([𝐢] ∼ β‰ˆ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β†’ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β‰ˆ [𝐢] ∼ )
106103, 104, 1053syl 18 . . . 4 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β‰ˆ [𝐢] ∼ )
107203ad2ant1 1133 . . . . . . 7 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ 𝐾 ∈ (SubGrpβ€˜πΊ))
10821, 2eqgen 19063 . . . . . . 7 ((𝐾 ∈ (SubGrpβ€˜πΊ) ∧ [𝐢] ∼ ∈ (𝑋 / ∼ )) β†’ 𝐾 β‰ˆ [𝐢] ∼ )
109107, 6, 108syl2anc 584 . . . . . 6 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ 𝐾 β‰ˆ [𝐢] ∼ )
110 ensym 9001 . . . . . 6 (𝐾 β‰ˆ [𝐢] ∼ β†’ [𝐢] ∼ β‰ˆ 𝐾)
111109, 110syl 17 . . . . 5 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ [𝐢] ∼ β‰ˆ 𝐾)
112 ecelqsg 8768 . . . . . . 7 (( ∼ ∈ V ∧ (𝐡 + 𝐢) ∈ 𝑋) β†’ [(𝐡 + 𝐢)] ∼ ∈ (𝑋 / ∼ ))
1133, 41, 112sylancr 587 . . . . . 6 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ [(𝐡 + 𝐢)] ∼ ∈ (𝑋 / ∼ ))
11421, 2eqgen 19063 . . . . . 6 ((𝐾 ∈ (SubGrpβ€˜πΊ) ∧ [(𝐡 + 𝐢)] ∼ ∈ (𝑋 / ∼ )) β†’ 𝐾 β‰ˆ [(𝐡 + 𝐢)] ∼ )
115107, 113, 114syl2anc 584 . . . . 5 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ 𝐾 β‰ˆ [(𝐡 + 𝐢)] ∼ )
116 entr 9004 . . . . 5 (([𝐢] ∼ β‰ˆ 𝐾 ∧ 𝐾 β‰ˆ [(𝐡 + 𝐢)] ∼ ) β†’ [𝐢] ∼ β‰ˆ [(𝐡 + 𝐢)] ∼ )
117111, 115, 116syl2anc 584 . . . 4 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ [𝐢] ∼ β‰ˆ [(𝐡 + 𝐢)] ∼ )
118 entr 9004 . . . 4 ((ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β‰ˆ [𝐢] ∼ ∧ [𝐢] ∼ β‰ˆ [(𝐡 + 𝐢)] ∼ ) β†’ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β‰ˆ [(𝐡 + 𝐢)] ∼ )
119106, 117, 118syl2anc 584 . . 3 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β‰ˆ [(𝐡 + 𝐢)] ∼ )
120 fisseneq 9259 . . 3 (([(𝐡 + 𝐢)] ∼ ∈ Fin ∧ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) βŠ† [(𝐡 + 𝐢)] ∼ ∧ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) β‰ˆ [(𝐡 + 𝐢)] ∼ ) β†’ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) = [(𝐡 + 𝐢)] ∼ )
12126, 88, 119, 120syl3anc 1371 . 2 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ ran (𝑧 ∈ [𝐢] ∼ ↦ (𝐡 + 𝑧)) = [(𝐡 + 𝐢)] ∼ )
12218, 121eqtrd 2772 1 ((πœ‘ ∧ 𝐡 ∈ 𝐻 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡 Β· [𝐢] ∼ ) = [(𝐡 + 𝐢)] ∼ )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677   β†Ύ cres 5678  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413   Er wer 8702  [cec 8703   / cqs 8704   β‰ˆ cen 8938  Fincfn 8941  Basecbs 17146  +gcplusg 17199  Grpcgrp 18821  invgcminusg 18822  -gcsg 18823  SubGrpcsubg 19002   ~QG cqg 19004
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-ec 8707  df-qs 8711  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824  df-minusg 18825  df-sbg 18826  df-subg 19005  df-eqg 19007
This theorem is referenced by:  sylow2blem2  19491  sylow2blem3  19492
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