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Theorem slwhash 19491
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 19477 . . . . 5 (𝐻 ∈ (𝑃 pSyl 𝐺) β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
42, 3syl 17 . . . 4 (πœ‘ β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
5 subgrcl 19010 . . . 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 19476 . . . 4 (𝐻 ∈ (𝑃 pSyl 𝐺) β†’ 𝑃 ∈ β„™)
92, 8syl 17 . . 3 (πœ‘ β†’ 𝑃 ∈ β„™)
101grpbn0 18850 . . . . . 6 (𝐺 ∈ Grp β†’ 𝑋 β‰  βˆ…)
116, 10syl 17 . . . . 5 (πœ‘ β†’ 𝑋 β‰  βˆ…)
12 hashnncl 14325 . . . . . 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 256 . . . 4 (πœ‘ β†’ (β™―β€˜π‘‹) ∈ β„•)
159, 14pccld 16782 . . 3 (πœ‘ β†’ (𝑃 pCnt (β™―β€˜π‘‹)) ∈ β„•0)
16 pcdvds 16796 . . . 4 ((𝑃 ∈ β„™ ∧ (β™―β€˜π‘‹) ∈ β„•) β†’ (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))) βˆ₯ (β™―β€˜π‘‹))
179, 14, 16syl2anc 584 . . 3 (πœ‘ β†’ (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))) βˆ₯ (β™―β€˜π‘‹))
181, 6, 7, 9, 15, 17sylow1 19470 . 2 (πœ‘ β†’ βˆƒπ‘˜ ∈ (SubGrpβ€˜πΊ)(β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))
197adantr 481 . . . 4 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ 𝑋 ∈ Fin)
204adantr 481 . . . 4 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
21 simprl 769 . . . 4 ((πœ‘ ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ (β™―β€˜π‘˜) = (𝑃↑(𝑃 pCnt (β™―β€˜π‘‹))))) β†’ π‘˜ ∈ (SubGrpβ€˜πΊ))
22 eqid 2732 . . . 4 (+gβ€˜πΊ) = (+gβ€˜πΊ)
23 eqid 2732 . . . . . . 7 (𝐺 β†Ύs 𝐻) = (𝐺 β†Ύs 𝐻)
2423slwpgp 19480 . . . . . 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 481 . . . 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 771 . . . 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 2732 . . . 4 (-gβ€˜πΊ) = (-gβ€˜πΊ)
291, 19, 20, 21, 22, 26, 27, 28sylow2b 19490 . . 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 771 . . . . . 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 724 . . . . . . . 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 724 . . . . . . . 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 481 . . . . . . . . . . . 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 769 . . . . . . . . . . . 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 2732 . . . . . . . . . . . . 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 19123 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 2732 . . . . . . . . . . . 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 19009 . . . . . . . . . . 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 19124 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 724 . . . . . . . . . . . . 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 19006 . . . . . . . . . . . . . 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 9266 . . . . . . . . . . . 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 19006 . . . . . . . . . . . . . 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 9266 . . . . . . . . . . . 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 14306 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 256 . . . . . . . . . 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 776 . . . . . . . . . 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 2774 . . . . . . . . 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 2774 . . . . . . . 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 7416 . . . . . . . . 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 584 . . . . . . 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 19008 . . . . . . . . 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 2834 . . . . . . . 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 2732 . . . . . . . . 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 19472 . . . . . . . 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 584 . . . . . . 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 711 . . . . . 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 19478 . . . . . . 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 584 . . . . . 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 710 . . . . 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 2772 . . 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 3161 . 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 3161 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   βŠ† wss 3948  βˆ…c0 4322   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  (class class class)co 7408   β‰ˆ cen 8935  Fincfn 8938  β„•cn 12211  β„•0cn0 12471  β†‘cexp 14026  β™―chash 14289   βˆ₯ cdvds 16196  β„™cprime 16607   pCnt cpc 16768  Basecbs 17143   β†Ύs cress 17172  +gcplusg 17196  Grpcgrp 18818  -gcsg 18820  SubGrpcsubg 18999   pGrp cpgp 19393   pSyl cslw 19394
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 7724  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
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-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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-2o 8466  df-oadd 8469  df-omul 8470  df-er 8702  df-ec 8704  df-qs 8708  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-oi 9504  df-dju 9895  df-card 9933  df-acn 9936  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-xnn0 12544  df-z 12558  df-uz 12822  df-q 12932  df-rp 12974  df-fz 13484  df-fzo 13627  df-fl 13756  df-mod 13834  df-seq 13966  df-exp 14027  df-fac 14233  df-bc 14262  df-hash 14290  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-clim 15431  df-sum 15632  df-dvds 16197  df-gcd 16435  df-prm 16608  df-pc 16769  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-submnd 18671  df-grp 18821  df-minusg 18822  df-sbg 18823  df-mulg 18950  df-subg 19002  df-eqg 19004  df-ghm 19089  df-ga 19153  df-od 19395  df-pgp 19397  df-slw 19398
This theorem is referenced by:  fislw  19492  sylow2  19493  sylow3lem4  19497
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