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Mirrors > Home > MPE Home > Th. List > sylow3lem5 | Structured version Visualization version GIF version |
Description: Lemma for sylow3 19631, second part. Reduce the group action of sylow3lem1 19625 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 19608 | . . . . . 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 19121 | . . . . 5 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ⊆ 𝑋) |
7 | ssid 4002 | . . . 4 ⊢ (𝑃 pSyl 𝐺) ⊆ (𝑃 pSyl 𝐺) | |
8 | resmpo 7545 | . . . 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 584 | . . 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 | eqtr4di 2784 | . 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 7432 | . . . . . . . . 9 ⊢ (𝑧 = 𝑐 → (𝑥 + 𝑧) = (𝑥 + 𝑐)) | |
18 | 17 | oveq1d 7439 | . . . . . . . 8 ⊢ (𝑧 = 𝑐 → ((𝑥 + 𝑧) − 𝑥) = ((𝑥 + 𝑐) − 𝑥)) |
19 | 18 | cbvmptv 5266 | . . . . . . 7 ⊢ (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑐 ∈ 𝑦 ↦ ((𝑥 + 𝑐) − 𝑥)) |
20 | oveq1 7431 | . . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 + 𝑐) = (𝑎 + 𝑐)) | |
21 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎) | |
22 | 20, 21 | oveq12d 7442 | . . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥 + 𝑐) − 𝑥) = ((𝑎 + 𝑐) − 𝑎)) |
23 | 22 | mpteq2dv 5255 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑐 ∈ 𝑦 ↦ ((𝑥 + 𝑐) − 𝑥)) = (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎))) |
24 | 19, 23 | eqtrid 2778 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎))) |
25 | 24 | rneqd 5944 | . . . . 5 ⊢ (𝑥 = 𝑎 → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎))) |
26 | mpteq1 5246 | . . . . . 6 ⊢ (𝑦 = 𝑏 → (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎)) = (𝑐 ∈ 𝑏 ↦ ((𝑎 + 𝑐) − 𝑎))) | |
27 | 26 | rneqd 5944 | . . . . 5 ⊢ (𝑦 = 𝑏 → ran (𝑐 ∈ 𝑦 ↦ ((𝑎 + 𝑐) − 𝑎)) = ran (𝑐 ∈ 𝑏 ↦ ((𝑎 + 𝑐) − 𝑎))) |
28 | 25, 27 | cbvmpov 7520 | . . . 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 19625 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺))) |
30 | eqid 2726 | . . . 4 ⊢ (𝐺 ↾s 𝐾) = (𝐺 ↾s 𝐾) | |
31 | 30 | gasubg 19296 | . . 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 582 | . 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 2827 | 1 ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾s 𝐾) GrpAct (𝑃 pSyl 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ↦ cmpt 5236 × cxp 5680 ran crn 5683 ↾ cres 5684 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 Fincfn 8974 ℙcprime 16672 Basecbs 17213 ↾s cress 17242 +gcplusg 17266 Grpcgrp 18928 -gcsg 18930 SubGrpcsubg 19114 GrpAct cga 19283 pSyl cslw 19525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-disj 5119 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-omul 8501 df-er 8734 df-ec 8736 df-qs 8740 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-acn 9985 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12597 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-sum 15691 df-dvds 16257 df-gcd 16495 df-prm 16673 df-pc 16839 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-eqg 19119 df-ghm 19207 df-ga 19284 df-od 19526 df-pgp 19528 df-slw 19529 |
This theorem is referenced by: sylow3lem6 19630 |
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