sylow2 - Metamath Proof Explorer

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Theorem sylow2 19612

Description: Sylow's second theorem. See also sylow2b 19609 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 19611). 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 19596	. . . . . . 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 19238	. . . . . 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 19115	. . . . 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 9278	. . 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 19610	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
19	7, 1, 3	slwhash 19610	. . . . . 6 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
20	18, 19	eqtr4d 2774	. . . . 5 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝐾))
21		slwsubg 19596	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
22	17, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
23	7	subgss 19115	. . . . . . . 8 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
25	1, 24	ssfid 9278	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Fin)
26	7	subgss 19115	. . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
27	5, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
28	1, 27	ssfid 9278	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin)
29		hashen 14370	. . . . . 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 19239	. . . . 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 9025	. . . 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 9270	. . 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 1373	. 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 19599	. . . 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 19609	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
42	37, 41	reximddv 3157	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 1540 ∈ wcel 2109 ∃wrex 3061 ⊆ wss 3931 class class class wbr 5124 ↦ cmpt 5206 ran crn 5660 ‘cfv 6536 (class class class)co 7410 ≈ cen 8961 Fincfn 8964 ↑cexp 14084 ♯chash 14353 pCnt cpc 16861 Basecbs 17233 ↾_s cress 17256 +_gcplusg 17276 -_gcsg 18923 SubGrpcsubg 19108 pGrp cpgp 19512 pSyl cslw 19513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12858 df-q 12970 df-rp 13014 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-sum 15708 df-dvds 16278 df-gcd 16519 df-prm 16696 df-pc 16862 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-0g 17460 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-eqg 19113 df-ghm 19201 df-ga 19278 df-od 19514 df-pgp 19516 df-slw 19517

This theorem is referenced by: sylow3lem3 19615 sylow3lem6 19618

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