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Mirrors > Home > MPE Home > Th. List > sylow3lem5 | Structured version Visualization version GIF version |
Description: Lemma for sylow3 18752, second part. Reduce the group action of sylow3lem1 18746 to a given Sylow subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.) |
Ref | Expression |
---|---|
sylow3.x | ⊢ 𝑋 = (Base‘𝐺) |
sylow3.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
sylow3.xf | ⊢ (𝜑 → 𝑋 ∈ Fin) |
sylow3.p | ⊢ (𝜑 → 𝑃 ∈ ℙ) |
sylow3lem5.a | ⊢ + = (+g‘𝐺) |
sylow3lem5.d | ⊢ − = (-g‘𝐺) |
sylow3lem5.k | ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) |
sylow3lem5.m | ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) |
Ref | Expression |
---|---|
sylow3lem5 | ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾s 𝐾) GrpAct (𝑃 pSyl 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylow3lem5.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) | |
2 | slwsubg 18729 | . . . . . 6 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺)) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
4 | sylow3.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
5 | 4 | subgss 18274 | . . . . 5 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ⊆ 𝑋) |
7 | ssid 3988 | . . . 4 ⊢ (𝑃 pSyl 𝐺) ⊆ (𝑃 pSyl 𝐺) | |
8 | resmpo 7266 | . . . 4 ⊢ ((𝐾 ⊆ 𝑋 ∧ (𝑃 pSyl 𝐺) ⊆ (𝑃 pSyl 𝐺)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ↾ (𝐾 × (𝑃 pSyl 𝐺))) = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))) | |
9 | 6, 7, 8 | sylancl 588 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ↾ (𝐾 × (𝑃 pSyl 𝐺))) = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))) |
10 | sylow3lem5.m | . . 3 ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) | |
11 | 9, 10 | syl6eqr 2874 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ↾ (𝐾 × (𝑃 pSyl 𝐺))) = ⊕ ) |
12 | sylow3.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
13 | sylow3.xf | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
14 | sylow3.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ) | |
15 | sylow3lem5.a | . . . 4 ⊢ + = (+g‘𝐺) | |
16 | sylow3lem5.d | . . . 4 ⊢ − = (-g‘𝐺) | |
17 | oveq2 7158 | . . . . . . . . 9 ⊢ (𝑧 = 𝑐 → (𝑥 + 𝑧) = (𝑥 + 𝑐)) | |
18 | 17 | oveq1d 7165 | . . . . . . . 8 ⊢ (𝑧 = 𝑐 → ((𝑥 + 𝑧) − 𝑥) = ((𝑥 + 𝑐) − 𝑥)) |
19 | 18 | cbvmptv 5161 | . . . . . . 7 ⊢ (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑐 ∈ 𝑦 ↦ ((𝑥 + 𝑐) − 𝑥)) |
20 | oveq1 7157 | . . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 + 𝑐) = (𝑎 + 𝑐)) | |
21 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎) | |
22 | 20, 21 | oveq12d 7168 | . . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥 + 𝑐) − 𝑥) = ((𝑎 + 𝑐) − 𝑎)) |
23 | 22 | mpteq2dv 5154 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑐 ∈ 𝑦 ↦ ((𝑥 + 𝑐) − 𝑥)) = (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎))) |
24 | 19, 23 | syl5eq 2868 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎))) |
25 | 24 | rneqd 5802 | . . . . 5 ⊢ (𝑥 = 𝑎 → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎))) |
26 | mpteq1 5146 | . . . . . 6 ⊢ (𝑦 = 𝑏 → (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎)) = (𝑐 ∈ 𝑏 ↦ ((𝑎 + 𝑐) − 𝑎))) | |
27 | 26 | rneqd 5802 | . . . . 5 ⊢ (𝑦 = 𝑏 → ran (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎)) = ran (𝑐 ∈ 𝑏 ↦ ((𝑎 + 𝑐) − 𝑎))) |
28 | 25, 27 | cbvmpov 7243 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) = (𝑎 ∈ 𝑋, 𝑏 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑐 ∈ 𝑏 ↦ ((𝑎 + 𝑐) − 𝑎))) |
29 | 4, 12, 13, 14, 15, 16, 28 | sylow3lem1 18746 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺))) |
30 | eqid 2821 | . . . 4 ⊢ (𝐺 ↾s 𝐾) = (𝐺 ↾s 𝐾) | |
31 | 30 | gasubg 18426 | . . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ↾ (𝐾 × (𝑃 pSyl 𝐺))) ∈ ((𝐺 ↾s 𝐾) GrpAct (𝑃 pSyl 𝐺))) |
32 | 29, 3, 31 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ↾ (𝐾 × (𝑃 pSyl 𝐺))) ∈ ((𝐺 ↾s 𝐾) GrpAct (𝑃 pSyl 𝐺))) |
33 | 11, 32 | eqeltrrd 2914 | 1 ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾s 𝐾) GrpAct (𝑃 pSyl 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ↦ cmpt 5138 × cxp 5547 ran crn 5550 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 Fincfn 8503 ℙcprime 16009 Basecbs 16477 ↾s cress 16478 +gcplusg 16559 Grpcgrp 18097 -gcsg 18099 SubGrpcsubg 18267 GrpAct cga 18413 pSyl cslw 18649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-dvds 15602 df-gcd 15838 df-prm 16010 df-pc 16168 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-eqg 18272 df-ghm 18350 df-ga 18414 df-od 18650 df-pgp 18652 df-slw 18653 |
This theorem is referenced by: sylow3lem6 18751 |
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