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Mirrors > Home > MPE Home > Th. List > cygth | Structured version Visualization version GIF version |
Description: The "fundamental theorem of cyclic groups". Cyclic groups are exactly the additive groups ℤ / 𝑛ℤ, for 0 ≤ 𝑛 (where 𝑛 = 0 is the infinite cyclic group ℤ), up to isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
cygth | ⊢ (𝐺 ∈ CycGrp ↔ ∃𝑛 ∈ ℕ0 𝐺 ≃𝑔 (ℤ/nℤ‘𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 14185 | . . . . 5 ⊢ ((Base‘𝐺) ∈ Fin → (♯‘(Base‘𝐺)) ∈ ℕ0) | |
2 | 1 | adantl 483 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ (Base‘𝐺) ∈ Fin) → (♯‘(Base‘𝐺)) ∈ ℕ0) |
3 | 0nn0 12362 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ ¬ (Base‘𝐺) ∈ Fin) → 0 ∈ ℕ0) |
5 | 2, 4 | ifclda 4520 | . . 3 ⊢ (𝐺 ∈ CycGrp → if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) ∈ ℕ0) |
6 | eqid 2738 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
7 | eqid 2738 | . . . 4 ⊢ if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) = if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) | |
8 | eqid 2738 | . . . 4 ⊢ (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0)) = (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0)) | |
9 | 6, 7, 8 | cygzn 20906 | . . 3 ⊢ (𝐺 ∈ CycGrp → 𝐺 ≃𝑔 (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0))) |
10 | fveq2 6838 | . . . . 5 ⊢ (𝑛 = if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) → (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0))) | |
11 | 10 | breq2d 5116 | . . . 4 ⊢ (𝑛 = if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) → (𝐺 ≃𝑔 (ℤ/nℤ‘𝑛) ↔ 𝐺 ≃𝑔 (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0)))) |
12 | 11 | rspcev 3580 | . . 3 ⊢ ((if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) ∈ ℕ0 ∧ 𝐺 ≃𝑔 (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0))) → ∃𝑛 ∈ ℕ0 𝐺 ≃𝑔 (ℤ/nℤ‘𝑛)) |
13 | 5, 9, 12 | syl2anc 585 | . 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑛 ∈ ℕ0 𝐺 ≃𝑔 (ℤ/nℤ‘𝑛)) |
14 | gicsym 18999 | . . . 4 ⊢ (𝐺 ≃𝑔 (ℤ/nℤ‘𝑛) → (ℤ/nℤ‘𝑛) ≃𝑔 𝐺) | |
15 | eqid 2738 | . . . . 5 ⊢ (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘𝑛) | |
16 | 15 | zncyg 20884 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → (ℤ/nℤ‘𝑛) ∈ CycGrp) |
17 | giccyg 19612 | . . . 4 ⊢ ((ℤ/nℤ‘𝑛) ≃𝑔 𝐺 → ((ℤ/nℤ‘𝑛) ∈ CycGrp → 𝐺 ∈ CycGrp)) | |
18 | 14, 16, 17 | syl2imc 41 | . . 3 ⊢ (𝑛 ∈ ℕ0 → (𝐺 ≃𝑔 (ℤ/nℤ‘𝑛) → 𝐺 ∈ CycGrp)) |
19 | 18 | rexlimiv 3144 | . 2 ⊢ (∃𝑛 ∈ ℕ0 𝐺 ≃𝑔 (ℤ/nℤ‘𝑛) → 𝐺 ∈ CycGrp) |
20 | 13, 19 | impbii 208 | 1 ⊢ (𝐺 ∈ CycGrp ↔ ∃𝑛 ∈ ℕ0 𝐺 ≃𝑔 (ℤ/nℤ‘𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 ifcif 4485 class class class wbr 5104 ‘cfv 6492 Fincfn 8817 0cc0 10985 ℕ0cn0 12347 ♯chash 14159 Basecbs 17019 ≃𝑔 cgic 18983 CycGrpccyg 19589 ℤ/nℤczn 20832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-addf 11064 ax-mulf 11065 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-tpos 8125 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-oadd 8384 df-omul 8385 df-er 8582 df-ec 8584 df-qs 8588 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-sup 9312 df-inf 9313 df-oi 9380 df-card 9809 df-acn 9812 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12553 df-uz 12698 df-rp 12846 df-fz 13355 df-fl 13627 df-mod 13705 df-seq 13837 df-exp 13898 df-hash 14160 df-cj 14919 df-re 14920 df-im 14921 df-sqrt 15055 df-abs 15056 df-dvds 16073 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-starv 17084 df-sca 17085 df-vsca 17086 df-ip 17087 df-tset 17088 df-ple 17089 df-ds 17091 df-unif 17092 df-0g 17259 df-imas 17326 df-qus 17327 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-mhm 18537 df-grp 18687 df-minusg 18688 df-sbg 18689 df-mulg 18808 df-subg 18860 df-nsg 18861 df-eqg 18862 df-ghm 18941 df-gim 18984 df-gic 18985 df-od 19245 df-cmn 19499 df-abl 19500 df-cyg 19590 df-mgp 19832 df-ur 19849 df-ring 19896 df-cring 19897 df-oppr 19978 df-dvdsr 19999 df-rnghom 20075 df-subrg 20149 df-lmod 20253 df-lss 20322 df-lsp 20362 df-sra 20562 df-rgmod 20563 df-lidl 20564 df-rsp 20565 df-2idl 20631 df-cnfld 20726 df-zring 20799 df-zrh 20833 df-zn 20836 |
This theorem is referenced by: (None) |
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