![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > odhash | Structured version Visualization version GIF version |
Description: An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
odhash.x | ⊢ 𝑋 = (Base‘𝐺) |
odhash.o | ⊢ 𝑂 = (od‘𝐺) |
odhash.k | ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) |
Ref | Expression |
---|---|
odhash | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (♯‘(𝐾‘{𝐴})) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 11737 | . . 3 ⊢ ℤ ∈ V | |
2 | ominf 8460 | . . . 4 ⊢ ¬ ω ∈ Fin | |
3 | znnen 15345 | . . . . . 6 ⊢ ℤ ≈ ℕ | |
4 | nnenom 13098 | . . . . . 6 ⊢ ℕ ≈ ω | |
5 | 3, 4 | entri 8295 | . . . . 5 ⊢ ℤ ≈ ω |
6 | enfi 8464 | . . . . 5 ⊢ (ℤ ≈ ω → (ℤ ∈ Fin ↔ ω ∈ Fin)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ (ℤ ∈ Fin ↔ ω ∈ Fin) |
8 | 2, 7 | mtbir 315 | . . 3 ⊢ ¬ ℤ ∈ Fin |
9 | hashinf 13440 | . . 3 ⊢ ((ℤ ∈ V ∧ ¬ ℤ ∈ Fin) → (♯‘ℤ) = +∞) | |
10 | 1, 8, 9 | mp2an 682 | . 2 ⊢ (♯‘ℤ) = +∞ |
11 | odhash.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
12 | eqid 2777 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
13 | odhash.o | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
14 | odhash.k | . . . 4 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
15 | 11, 12, 13, 14 | odf1o1 18371 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)):ℤ–1-1-onto→(𝐾‘{𝐴})) |
16 | 1 | f1oen 8262 | . . 3 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)):ℤ–1-1-onto→(𝐾‘{𝐴}) → ℤ ≈ (𝐾‘{𝐴})) |
17 | hasheni 13453 | . . 3 ⊢ (ℤ ≈ (𝐾‘{𝐴}) → (♯‘ℤ) = (♯‘(𝐾‘{𝐴}))) | |
18 | 15, 16, 17 | 3syl 18 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (♯‘ℤ) = (♯‘(𝐾‘{𝐴}))) |
19 | 10, 18 | syl5reqr 2828 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (♯‘(𝐾‘{𝐴})) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 Vcvv 3397 {csn 4397 class class class wbr 4886 ↦ cmpt 4965 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 ωcom 7343 ≈ cen 8238 Fincfn 8241 0cc0 10272 +∞cpnf 10408 ℕcn 11374 ℤcz 11728 ♯chash 13435 Basecbs 16255 mrClscmrc 16629 Grpcgrp 17809 .gcmg 17927 SubGrpcsubg 17972 odcod 18328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-omul 7848 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fz 12644 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-dvds 15388 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-od 18332 |
This theorem is referenced by: odhash3 18375 |
Copyright terms: Public domain | W3C validator |