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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 16618 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
2 | 1 | adantl 481 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℕ) |
3 | 2 | nncnd 12235 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℂ) |
4 | 3 | exp0d 14112 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (𝑃↑0) = 1) |
5 | pgp0.1 | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
6 | 5 | fvexi 6905 | . . . . 5 ⊢ 0 ∈ V |
7 | hashsng 14336 | . . . . 5 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ (♯‘{ 0 }) = 1 |
9 | 5 | 0subg 19071 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
10 | 9 | adantr 480 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → { 0 } ∈ (SubGrp‘𝐺)) |
11 | eqid 2731 | . . . . . . 7 ⊢ (𝐺 ↾s { 0 }) = (𝐺 ↾s { 0 }) | |
12 | 11 | subgbas 19050 | . . . . . 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 6895 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (♯‘{ 0 }) = (♯‘(Base‘(𝐺 ↾s { 0 })))) |
15 | 8, 14 | eqtr3id 2785 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 1 = (♯‘(Base‘(𝐺 ↾s { 0 })))) |
16 | 4, 15 | eqtr2d 2772 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (♯‘(Base‘(𝐺 ↾s { 0 }))) = (𝑃↑0)) |
17 | 11 | subggrp 19049 | . . . 4 ⊢ ({ 0 } ∈ (SubGrp‘𝐺) → (𝐺 ↾s { 0 }) ∈ Grp) |
18 | 10, 17 | syl 17 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (𝐺 ↾s { 0 }) ∈ Grp) |
19 | simpr 484 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℙ) | |
20 | 0nn0 12494 | . . . 4 ⊢ 0 ∈ ℕ0 | |
21 | 20 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 0 ∈ ℕ0) |
22 | eqid 2731 | . . . 4 ⊢ (Base‘(𝐺 ↾s { 0 })) = (Base‘(𝐺 ↾s { 0 })) | |
23 | 22 | pgpfi1 19508 | . . 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 1370 | . 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 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 {csn 4628 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 0cc0 11116 1c1 11117 ℕcn 12219 ℕ0cn0 12479 ↑cexp 14034 ♯chash 14297 ℙcprime 16615 Basecbs 17151 ↾s cress 17180 0gc0g 17392 Grpcgrp 18858 SubGrpcsubg 19040 pGrp cpgp 19439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-omul 8477 df-er 8709 df-ec 8711 df-qs 8715 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-acn 9943 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-dvds 16205 df-gcd 16443 df-prm 16616 df-pc 16777 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-subg 19043 df-eqg 19045 df-od 19441 df-pgp 19443 |
This theorem is referenced by: slwn0 19528 |
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