sylow2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sylow2		Structured version Visualization version GIF version

Theorem sylow2 19645

Description: Sylow's second theorem. See also sylow2b 19642 for the "hard" part of the proof. Any two Sylow 𝑃-subgroups are conjugate to one another, and hence the same size, namely 𝑃↑(𝑃 pCnt ∣ 𝑋 ∣ ) (see fislw 19644). This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)

**Hypotheses**
Ref	Expression
sylow2.x	⊢ 𝑋 = (Base‘𝐺)
sylow2.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow2.h	⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺))
sylow2.k	⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
sylow2.a	⊢ + = (+_g‘𝐺)
sylow2.d	⊢ − = (-_g‘𝐺)

**Assertion**
Ref	Expression
sylow2	⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))

Distinct variable groups: 𝑥, − 𝑥,𝑔, + 𝑔,𝐺,𝑥 𝑔,𝐻,𝑥 𝑔,𝐾,𝑥 𝜑,𝑔 𝑔,𝑋,𝑥 Allowed substitution hints: 𝜑(𝑥) 𝑃(𝑥,𝑔) − (𝑔)

**Proof of Theorem sylow2**
Step	Hyp	Ref	Expression
1		sylow2.f	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝑋 ∈ Fin)
3		sylow2.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
4		slwsubg 19629	. . . . . . 7 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
6		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝑔 ∈ 𝑋)
7		sylow2.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
8		sylow2.a	. . . . . . 7 ⊢ + = (+_g‘𝐺)
9		sylow2.d	. . . . . . 7 ⊢ − = (-_g‘𝐺)
10		eqid 2736	. . . . . . 7 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) = (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))
11	7, 8, 9, 10	conjsubg 19269	. . . . . 6 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺))
12	5, 6, 11	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺))
13	7	subgss 19146	. . . . 5 ⊢ (ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ⊆ 𝑋)
14	12, 13	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ⊆ 𝑋)
15	2, 14	ssfid 9302	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ Fin)
16		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
17		sylow2.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺))
18	7, 1, 17	slwhash 19643	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
19	7, 1, 3	slwhash 19643	. . . . . 6 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
20	18, 19	eqtr4d 2779	. . . . 5 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝐾))
21		slwsubg 19629	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
22	17, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
23	7	subgss 19146	. . . . . . . 8 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
25	1, 24	ssfid 9302	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Fin)
26	7	subgss 19146	. . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
27	5, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
28	1, 27	ssfid 9302	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin)
29		hashen 14387	. . . . . 6 ⊢ ((𝐻 ∈ Fin ∧ 𝐾 ∈ Fin) → ((♯‘𝐻) = (♯‘𝐾) ↔ 𝐻 ≈ 𝐾))
30	25, 28, 29	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((♯‘𝐻) = (♯‘𝐾) ↔ 𝐻 ≈ 𝐾))
31	20, 30	mpbid 232	. . . 4 ⊢ (𝜑 → 𝐻 ≈ 𝐾)
32	7, 8, 9, 10	conjsubgen 19270	. . . . 5 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
33	5, 6, 32	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
34		entr 9047	. . . 4 ⊢ ((𝐻 ≈ 𝐾 ∧ 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
35	31, 33, 34	syl2an2r 685	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
36		fisseneq 9294	. . 3 ⊢ ((ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ Fin ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∧ 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) → 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
37	15, 16, 35, 36	syl3anc 1372	. 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
38		eqid 2736	. . . . 5 ⊢ (𝐺 ↾_s 𝐻) = (𝐺 ↾_s 𝐻)
39	38	slwpgp 19632	. . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾_s 𝐻))
40	17, 39	syl 17	. . 3 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s 𝐻))
41	7, 1, 22, 5, 8, 40, 19, 9	sylow2b 19642	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
42	37, 41	reximddv 3170	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ⊆ wss 3950 class class class wbr 5142 ↦ cmpt 5224 ran crn 5685 ‘cfv 6560 (class class class)co 7432 ≈ cen 8983 Fincfn 8986 ↑cexp 14103 ♯chash 14370 pCnt cpc 16875 Basecbs 17248 ↾_s cress 17275 +_gcplusg 17298 -_gcsg 18954 SubGrpcsubg 19139 pGrp cpgp 19545 pSyl cslw 19546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-dvds 16292 df-gcd 16533 df-prm 16710 df-pc 16876 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-eqg 19144 df-ghm 19232 df-ga 19309 df-od 19547 df-pgp 19549 df-slw 19550

This theorem is referenced by: sylow3lem3 19648 sylow3lem6 19651

W3C validator