Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 14163 | . . . . 5 ⊢ ((Base‘𝐺) ∈ Fin → (♯‘(Base‘𝐺)) ∈ ℕ0) | |
2 | 1 | adantl 482 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ (Base‘𝐺) ∈ Fin) → (♯‘(Base‘𝐺)) ∈ ℕ0) |
3 | 0nn0 12341 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ ¬ (Base‘𝐺) ∈ Fin) → 0 ∈ ℕ0) |
5 | 2, 4 | ifclda 4507 | . . 3 ⊢ (𝐺 ∈ CycGrp → if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) ∈ ℕ0) |
6 | eqid 2736 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
7 | eqid 2736 | . . . 4 ⊢ if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) = if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) | |
8 | eqid 2736 | . . . 4 ⊢ (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0)) = (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0)) | |
9 | 6, 7, 8 | cygzn 20876 | . . 3 ⊢ (𝐺 ∈ CycGrp → 𝐺 ≃𝑔 (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0))) |
10 | fveq2 6819 | . . . . 5 ⊢ (𝑛 = if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0) → (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘if((Base‘𝐺) ∈ Fin, (♯‘(Base‘𝐺)), 0))) | |
11 | 10 | breq2d 5101 | . . . 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 3570 | . . 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 584 | . 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑛 ∈ ℕ0 𝐺 ≃𝑔 (ℤ/nℤ‘𝑛)) |
14 | gicsym 18978 | . . . 4 ⊢ (𝐺 ≃𝑔 (ℤ/nℤ‘𝑛) → (ℤ/nℤ‘𝑛) ≃𝑔 𝐺) | |
15 | eqid 2736 | . . . . 5 ⊢ (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘𝑛) | |
16 | 15 | zncyg 20854 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → (ℤ/nℤ‘𝑛) ∈ CycGrp) |
17 | giccyg 19588 | . . . 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 3141 | . 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 396 = wceq 1540 ∈ wcel 2105 ∃wrex 3070 ifcif 4472 class class class wbr 5089 ‘cfv 6473 Fincfn 8796 0cc0 10964 ℕ0cn0 12326 ♯chash 14137 Basecbs 17001 ≃𝑔 cgic 18962 CycGrpccyg 19564 ℤ/nℤczn 20802 |
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 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042 ax-addf 11043 ax-mulf 11044 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-tpos 8104 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-oadd 8363 df-omul 8364 df-er 8561 df-ec 8563 df-qs 8567 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-sup 9291 df-inf 9292 df-oi 9359 df-card 9788 df-acn 9791 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-rp 12824 df-fz 13333 df-fl 13605 df-mod 13683 df-seq 13815 df-exp 13876 df-hash 14138 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-dvds 16055 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-starv 17066 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-unif 17074 df-0g 17241 df-imas 17308 df-qus 17309 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-grp 18668 df-minusg 18669 df-sbg 18670 df-mulg 18789 df-subg 18840 df-nsg 18841 df-eqg 18842 df-ghm 18920 df-gim 18963 df-gic 18964 df-od 19224 df-cmn 19475 df-abl 19476 df-cyg 19565 df-mgp 19808 df-ur 19825 df-ring 19872 df-cring 19873 df-oppr 19949 df-dvdsr 19970 df-rnghom 20046 df-subrg 20119 df-lmod 20223 df-lss 20292 df-lsp 20332 df-sra 20532 df-rgmod 20533 df-lidl 20534 df-rsp 20535 df-2idl 20601 df-cnfld 20696 df-zring 20769 df-zrh 20803 df-zn 20806 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |