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Theorem slwhash 19540
Description: A sylow subgroup has cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
fislw.1 𝑋 = (Baseβ€˜πΊ)
slwhash.3 (πœ‘ β†’ 𝑋 ∈ Fin)
slwhash.4 (πœ‘ β†’ 𝐻 ∈ (𝑃 pSyl 𝐺))
Assertion
Ref Expression
slwhash (πœ‘ β†’ (β™―β€˜π») = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
Proof of Theorem slwhash
Dummy variables 𝑔 π‘˜ 𝑛 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fislw.1 . . 3 𝑋 = (Baseβ€˜πΊ)
2 slwhash.4 . . . . 5 (πœ‘ β†’ 𝐻 ∈ (𝑃 pSyl 𝐺))
3 slwsubg 19526 . . . . 5 (𝐻 ∈ (𝑃 pSyl 𝐺) β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
42, 3syl 17 . . . 4 (πœ‘ β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
5 subgrcl 19054 . . . 4 (𝐻 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
64, 5syl 17 . . 3 (πœ‘ β†’ 𝐺 ∈ Grp)
7 slwhash.3 . . 3 (πœ‘ β†’ 𝑋 ∈ Fin)
8 slwprm 19525 . . . 4 (𝐻 ∈ (𝑃 pSyl 𝐺) β†’ 𝑃 ∈ β„™)
92, 8syl 17 . . 3 (πœ‘ β†’ 𝑃 ∈ β„™)
101grpbn0 18894 . . . . . 6 (𝐺 ∈ Grp β†’ 𝑋 β‰  βˆ…)
116, 10syl 17 . . . . 5 (πœ‘ β†’ 𝑋 β‰  βˆ…)
12 hashnncl 14333 . . . . . 6 (𝑋 ∈ Fin β†’ ((β™―β€˜π‘‹) ∈ β„• ↔ 𝑋 β‰  βˆ…))
137, 12syl 17 . . . . 5 (πœ‘ β†’ ((β™―β€˜π‘‹) ∈ β„• ↔ 𝑋 β‰  βˆ…))
1411, 13mpbird 257 . . . 4 (πœ‘ β†’ (β™―β€˜π‘‹) ∈ β„•)
159, 14pccld 16790 . . 3 (πœ‘ β†’ (𝑃 pCnt (β™―β€˜π‘‹)) ∈ β„•0)
16 pcdvds 16804 . . . 4 ((𝑃 ∈ β„™ ∧ (β™―β€˜π‘‹) ∈ β„•) β†’ (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))) βˆ₯ (β™―β€˜π‘‹))
179, 14, 16syl2anc 583 . . 3 (πœ‘ β†’ (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))) βˆ₯ (β™―β€˜π‘‹))
181, 6, 7, 9, 15, 17sylow1 19519 . 2 (πœ‘ β†’ βˆƒπ‘˜ ∈ (SubGrpβ€˜πΊ)(β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
197adantr 480 . . . 4 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ 𝑋 ∈ Fin)
204adantr 480 . . . 4 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
21 simprl 768 . . . 4 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ π‘˜ ∈ (SubGrpβ€˜πΊ))
22 eqid 2731 . . . 4 (+gβ€˜πΊ) = (+gβ€˜πΊ)
23 eqid 2731 . . . . . . 7 (𝐺 β†Ύs 𝐻) = (𝐺 β†Ύs 𝐻)
2423slwpgp 19529 . . . . . 6 (𝐻 ∈ (𝑃 pSyl 𝐺) β†’ 𝑃 pGrp (𝐺 β†Ύs 𝐻))
252, 24syl 17 . . . . 5 (πœ‘ β†’ 𝑃 pGrp (𝐺 β†Ύs 𝐻))
2625adantr 480 . . . 4 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ 𝑃 pGrp (𝐺 β†Ύs 𝐻))
27 simprr 770 . . . 4 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
28 eqid 2731 . . . 4 (-gβ€˜πΊ) = (-gβ€˜πΊ)
291, 19, 20, 21, 22, 26, 27, 28sylow2b 19539 . . 3 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ βˆƒπ‘” ∈ 𝑋 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))
30 simprr 770 . . . . . 6 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))
312ad2antrr 723 . . . . . . . 8 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ 𝐻 ∈ (𝑃 pSyl 𝐺))
3231, 8syl 17 . . . . . . 7 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ 𝑃 ∈ β„™)
3315ad2antrr 723 . . . . . . . 8 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (𝑃 pCnt (β™―β€˜π‘‹)) ∈ β„•0)
3421adantr 480 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ π‘˜ ∈ (SubGrpβ€˜πΊ))
35 simprl 768 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ 𝑔 ∈ 𝑋)
36 eqid 2731 . . . . . . . . . . . . 13 (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) = (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))
371, 22, 28, 36conjsubg 19171 . . . . . . . . . . . 12 ((π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ 𝑔 ∈ 𝑋) β†’ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∈ (SubGrpβ€˜πΊ))
3834, 35, 37syl2anc 583 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∈ (SubGrpβ€˜πΊ))
39 eqid 2731 . . . . . . . . . . . 12 (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) = (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))
4039subgbas 19053 . . . . . . . . . . 11 (ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∈ (SubGrpβ€˜πΊ) β†’ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) = (Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))))
4138, 40syl 17 . . . . . . . . . 10 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) = (Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))))
4241fveq2d 6895 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (β™―β€˜ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) = (β™―β€˜(Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))))
431, 22, 28, 36conjsubgen 19172 . . . . . . . . . . . 12 ((π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ 𝑔 ∈ 𝑋) β†’ π‘˜ β‰ˆ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))
4434, 35, 43syl2anc 583 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ π‘˜ β‰ˆ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))
457ad2antrr 723 . . . . . . . . . . . . 13 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ 𝑋 ∈ Fin)
461subgss 19050 . . . . . . . . . . . . . 14 (π‘˜ ∈ (SubGrpβ€˜πΊ) β†’ π‘˜ βŠ† 𝑋)
4734, 46syl 17 . . . . . . . . . . . . 13 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ π‘˜ βŠ† 𝑋)
4845, 47ssfid 9273 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ π‘˜ ∈ Fin)
491subgss 19050 . . . . . . . . . . . . . 14 (ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∈ (SubGrpβ€˜πΊ) β†’ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) βŠ† 𝑋)
5038, 49syl 17 . . . . . . . . . . . . 13 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) βŠ† 𝑋)
5145, 50ssfid 9273 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∈ Fin)
52 hashen 14314 . . . . . . . . . . . 12 ((π‘˜ ∈ Fin ∧ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∈ Fin) β†’ ((β™―β€˜π‘˜) = (β™―β€˜ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) ↔ π‘˜ β‰ˆ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))
5348, 51, 52syl2anc 583 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ ((β™―β€˜π‘˜) = (β™―β€˜ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) ↔ π‘˜ β‰ˆ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))
5444, 53mpbird 257 . . . . . . . . . 10 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (β™―β€˜π‘˜) = (β™―β€˜ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))
55 simplrr 775 . . . . . . . . . 10 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
5654, 55eqtr3d 2773 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (β™―β€˜ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
5742, 56eqtr3d 2773 . . . . . . . 8 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (β™―β€˜(Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
58 oveq2 7420 . . . . . . . . 9 (𝑛 = (𝑃 pCnt (β™―β€˜π‘‹)) β†’ (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
5958rspceeqv 3633 . . . . . . . 8 (((𝑃 pCnt (β™―β€˜π‘‹)) ∈ β„•0 ∧ (β™―β€˜(Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹)))) β†’ βˆƒπ‘› ∈ β„•0 (β™―β€˜(Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))) = (𝑃↑𝑛))
6033, 57, 59syl2anc 583 . . . . . . 7 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ βˆƒπ‘› ∈ β„•0 (β™―β€˜(Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))) = (𝑃↑𝑛))
6139subggrp 19052 . . . . . . . . 9 (ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∈ (SubGrpβ€˜πΊ) β†’ (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) ∈ Grp)
6238, 61syl 17 . . . . . . . 8 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) ∈ Grp)
6341, 51eqeltrrd 2833 . . . . . . . 8 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) ∈ Fin)
64 eqid 2731 . . . . . . . . 9 (Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) = (Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))
6564pgpfi 19521 . . . . . . . 8 (((𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) ∈ Grp ∧ (Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) ∈ Fin) β†’ (𝑃 pGrp (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) ↔ (𝑃 ∈ β„™ ∧ βˆƒπ‘› ∈ β„•0 (β™―β€˜(Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))) = (𝑃↑𝑛))))
6662, 63, 65syl2anc 583 . . . . . . 7 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (𝑃 pGrp (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))) ↔ (𝑃 ∈ β„™ ∧ βˆƒπ‘› ∈ β„•0 (β™―β€˜(Baseβ€˜(𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))) = (𝑃↑𝑛))))
6732, 60, 66mpbir2and 710 . . . . . 6 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ 𝑃 pGrp (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))
6839slwispgp 19527 . . . . . . 7 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∈ (SubGrpβ€˜πΊ)) β†’ ((𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∧ 𝑃 pGrp (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) ↔ 𝐻 = ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))
6931, 38, 68syl2anc 583 . . . . . 6 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ ((𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)) ∧ 𝑃 pGrp (𝐺 β†Ύs ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) ↔ 𝐻 = ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))
7030, 67, 69mpbi2and 709 . . . . 5 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ 𝐻 = ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))
7170fveq2d 6895 . . . 4 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (β™―β€˜π») = (β™―β€˜ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔))))
7271, 56eqtrd 2771 . . 3 (((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 βŠ† ran (π‘₯ ∈ π‘˜ ↦ ((𝑔(+gβ€˜πΊ)π‘₯)(-gβ€˜πΊ)𝑔)))) β†’ (β™―β€˜π») = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
7329, 72rexlimddv 3160 . 2 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ (β™―β€˜π») = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
7418, 73rexlimddv 3160 1 (πœ‘ β†’ (β™―β€˜π») = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069   βŠ† wss 3948  βˆ…c0 4322   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  (class class class)co 7412   β‰ˆ cen 8942  Fincfn 8945  β„•cn 12219  β„•0cn0 12479  β†‘cexp 14034  β™―chash 14297   βˆ₯ cdvds 16204  β„™cprime 16615   pCnt cpc 16776  Basecbs 17151   β†Ύs cress 17180  +gcplusg 17204  Grpcgrp 18861  -gcsg 18863  SubGrpcsubg 19043   pGrp cpgp 19442   pSyl cslw 19443
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 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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-int 4951  df-iun 4999  df-disj 5114  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-se 5632  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-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-oadd 8476  df-omul 8477  df-er 8709  df-ec 8711  df-qs 8715  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-oi 9511  df-dju 9902  df-card 9940  df-acn 9943  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-xnn0 12552  df-z 12566  df-uz 12830  df-q 12940  df-rp 12982  df-fz 13492  df-fzo 13635  df-fl 13764  df-mod 13842  df-seq 13974  df-exp 14035  df-fac 14241  df-bc 14270  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640  df-dvds 16205  df-gcd 16443  df-prm 16616  df-pc 16777  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-0g 17394  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-submnd 18712  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18994  df-subg 19046  df-eqg 19048  df-ghm 19135  df-ga 19202  df-od 19444  df-pgp 19446  df-slw 19447
This theorem is referenced by:  fislw  19541  sylow2  19542  sylow3lem4  19546
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