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