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Mirrors > Home > MPE Home > Th. List > pgp0 | Structured version Visualization version GIF version |
Description: The identity subgroup is a 𝑃-group for every prime 𝑃. (Contributed by Mario Carneiro, 19-Jan-2015.) |
Ref | Expression |
---|---|
pgp0.1 | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
pgp0 | ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 pGrp (𝐺 ↾s { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmnn 16006 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
2 | 1 | adantl 482 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℕ) |
3 | 2 | nncnd 11642 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℂ) |
4 | 3 | exp0d 13492 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (𝑃↑0) = 1) |
5 | pgp0.1 | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
6 | 5 | fvexi 6677 | . . . . 5 ⊢ 0 ∈ V |
7 | hashsng 13718 | . . . . 5 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ (♯‘{ 0 }) = 1 |
9 | 5 | 0subg 18242 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
10 | 9 | adantr 481 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → { 0 } ∈ (SubGrp‘𝐺)) |
11 | eqid 2818 | . . . . . . 7 ⊢ (𝐺 ↾s { 0 }) = (𝐺 ↾s { 0 }) | |
12 | 11 | subgbas 18221 | . . . . . 6 ⊢ ({ 0 } ∈ (SubGrp‘𝐺) → { 0 } = (Base‘(𝐺 ↾s { 0 }))) |
13 | 10, 12 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → { 0 } = (Base‘(𝐺 ↾s { 0 }))) |
14 | 13 | fveq2d 6667 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (♯‘{ 0 }) = (♯‘(Base‘(𝐺 ↾s { 0 })))) |
15 | 8, 14 | syl5eqr 2867 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 1 = (♯‘(Base‘(𝐺 ↾s { 0 })))) |
16 | 4, 15 | eqtr2d 2854 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (♯‘(Base‘(𝐺 ↾s { 0 }))) = (𝑃↑0)) |
17 | 11 | subggrp 18220 | . . . 4 ⊢ ({ 0 } ∈ (SubGrp‘𝐺) → (𝐺 ↾s { 0 }) ∈ Grp) |
18 | 10, 17 | syl 17 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (𝐺 ↾s { 0 }) ∈ Grp) |
19 | simpr 485 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℙ) | |
20 | 0nn0 11900 | . . . 4 ⊢ 0 ∈ ℕ0 | |
21 | 20 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 0 ∈ ℕ0) |
22 | eqid 2818 | . . . 4 ⊢ (Base‘(𝐺 ↾s { 0 })) = (Base‘(𝐺 ↾s { 0 })) | |
23 | 22 | pgpfi1 18649 | . . 3 ⊢ (((𝐺 ↾s { 0 }) ∈ Grp ∧ 𝑃 ∈ ℙ ∧ 0 ∈ ℕ0) → ((♯‘(Base‘(𝐺 ↾s { 0 }))) = (𝑃↑0) → 𝑃 pGrp (𝐺 ↾s { 0 }))) |
24 | 18, 19, 21, 23 | syl3anc 1363 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → ((♯‘(Base‘(𝐺 ↾s { 0 }))) = (𝑃↑0) → 𝑃 pGrp (𝐺 ↾s { 0 }))) |
25 | 16, 24 | mpd 15 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 pGrp (𝐺 ↾s { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 ℕcn 11626 ℕ0cn0 11885 ↑cexp 13417 ♯chash 13678 ℙcprime 16003 Basecbs 16471 ↾s cress 16472 0gc0g 16701 Grpcgrp 18041 SubGrpcsubg 18211 pGrp cpgp 18583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-dvds 15596 df-gcd 15832 df-prm 16004 df-pc 16162 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-eqg 18216 df-od 18585 df-pgp 18587 |
This theorem is referenced by: slwn0 18669 |
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