sylow2 - Metamath Proof Explorer

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Theorem sylow2 19242

Description: Sylow's second theorem. See also sylow2b 19239 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 19241). 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 481	. . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝑋 ∈ Fin)
3		sylow2.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
4		slwsubg 19226	. . . . . . 7 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
6		simprl 768	. . . . . 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 2740	. . . . . . 7 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) = (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))
11	7, 8, 9, 10	conjsubg 18877	. . . . . 6 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺))
12	5, 6, 11	syl2an2r 682	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺))
13	7	subgss 18767	. . . . 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 9030	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ Fin)
16		simprr 770	. . 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 19240	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
19	7, 1, 3	slwhash 19240	. . . . . 6 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
20	18, 19	eqtr4d 2783	. . . . 5 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝐾))
21		slwsubg 19226	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
22	17, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
23	7	subgss 18767	. . . . . . . 8 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
25	1, 24	ssfid 9030	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Fin)
26	7	subgss 18767	. . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
27	5, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
28	1, 27	ssfid 9030	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin)
29		hashen 14072	. . . . . 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 231	. . . 4 ⊢ (𝜑 → 𝐻 ≈ 𝐾)
32	7, 8, 9, 10	conjsubgen 18878	. . . . 5 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
33	5, 6, 32	syl2an2r 682	. . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
34		entr 8784	. . . 4 ⊢ ((𝐻 ≈ 𝐾 ∧ 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
35	31, 33, 34	syl2an2r 682	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
36		fisseneq 9022	. . 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 1370	. 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
38		eqid 2740	. . . . 5 ⊢ (𝐺 ↾_s 𝐻) = (𝐺 ↾_s 𝐻)
39	38	slwpgp 19229	. . . 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 19239	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
42	37, 41	reximddv 3206	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 ⊆ wss 3892 class class class wbr 5079 ↦ cmpt 5162 ran crn 5591 ‘cfv 6432 (class class class)co 7272 ≈ cen 8722 Fincfn 8725 ↑cexp 13793 ♯chash 14055 pCnt cpc 16548 Basecbs 16923 ↾_s cress 16952 +_gcplusg 16973 -_gcsg 18590 SubGrpcsubg 18760 pGrp cpgp 19145 pSyl cslw 19146

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-disj 5045 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-oadd 8293 df-omul 8294 df-er 8490 df-ec 8492 df-qs 8496 df-map 8609 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-sup 9189 df-inf 9190 df-oi 9257 df-dju 9670 df-card 9708 df-acn 9711 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-n0 12245 df-xnn0 12317 df-z 12331 df-uz 12594 df-q 12700 df-rp 12742 df-fz 13251 df-fzo 13394 df-fl 13523 df-mod 13601 df-seq 13733 df-exp 13794 df-fac 13999 df-bc 14028 df-hash 14056 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-clim 15208 df-sum 15409 df-dvds 15975 df-gcd 16213 df-prm 16388 df-pc 16549 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-0g 17163 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-grp 18591 df-minusg 18592 df-sbg 18593 df-mulg 18712 df-subg 18763 df-eqg 18765 df-ghm 18843 df-ga 18907 df-od 19147 df-pgp 19149 df-slw 19150

This theorem is referenced by: sylow3lem3 19245 sylow3lem6 19248

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