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Mirrors > Home > MPE Home > Th. List > zncyg | Structured version Visualization version GIF version |
Description: The group ℤ / 𝑛ℤ is cyclic for all 𝑛 (including 𝑛 = 0). (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
zncyg.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
Ref | Expression |
---|---|
zncyg | ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CycGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zncyg.y | . . . . 5 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
2 | 1 | zncrng 20691 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
3 | crngring 19308 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Ring) |
5 | ringgrp 19302 | . . 3 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Grp) |
7 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
8 | eqid 2821 | . . . . 5 ⊢ (1r‘𝑌) = (1r‘𝑌) | |
9 | 7, 8 | ringidcl 19318 | . . . 4 ⊢ (𝑌 ∈ Ring → (1r‘𝑌) ∈ (Base‘𝑌)) |
10 | 4, 9 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1r‘𝑌) ∈ (Base‘𝑌)) |
11 | eqid 2821 | . . . . . . 7 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌) | |
12 | eqid 2821 | . . . . . . 7 ⊢ (.g‘𝑌) = (.g‘𝑌) | |
13 | 11, 12, 8 | zrhval2 20656 | . . . . . 6 ⊢ (𝑌 ∈ Ring → (ℤRHom‘𝑌) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
14 | 4, 13 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
15 | 14 | rneqd 5808 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ran (ℤRHom‘𝑌) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
16 | 1, 7, 11 | znzrhfo 20694 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌)) |
17 | forn 6593 | . . . . 5 ⊢ ((ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌) → ran (ℤRHom‘𝑌) = (Base‘𝑌)) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ran (ℤRHom‘𝑌) = (Base‘𝑌)) |
19 | 15, 18 | eqtr3d 2858 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌)) |
20 | oveq2 7164 | . . . . . . 7 ⊢ (𝑥 = (1r‘𝑌) → (𝑛(.g‘𝑌)𝑥) = (𝑛(.g‘𝑌)(1r‘𝑌))) | |
21 | 20 | mpteq2dv 5162 | . . . . . 6 ⊢ (𝑥 = (1r‘𝑌) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
22 | 21 | rneqd 5808 | . . . . 5 ⊢ (𝑥 = (1r‘𝑌) → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
23 | 22 | eqeq1d 2823 | . . . 4 ⊢ (𝑥 = (1r‘𝑌) → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌))) |
24 | 23 | rspcev 3623 | . . 3 ⊢ (((1r‘𝑌) ∈ (Base‘𝑌) ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌)) → ∃𝑥 ∈ (Base‘𝑌)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌)) |
25 | 10, 19, 24 | syl2anc 586 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑥 ∈ (Base‘𝑌)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌)) |
26 | 7, 12 | iscyg 18998 | . 2 ⊢ (𝑌 ∈ CycGrp ↔ (𝑌 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝑌)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌))) |
27 | 6, 25, 26 | sylanbrc 585 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CycGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ↦ cmpt 5146 ran crn 5556 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 ℕ0cn0 11898 ℤcz 11982 Basecbs 16483 Grpcgrp 18103 .gcmg 18224 CycGrpccyg 18996 1rcur 19251 Ringcrg 19297 CRingccrg 19298 ℤRHomczrh 20647 ℤ/nℤczn 20650 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-ec 8291 df-qs 8295 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-seq 13371 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-imas 16781 df-qus 16782 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-nsg 18277 df-eqg 18278 df-ghm 18356 df-cmn 18908 df-abl 18909 df-cyg 18997 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-rnghom 19467 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-sra 19944 df-rgmod 19945 df-lidl 19946 df-rsp 19947 df-2idl 20005 df-cnfld 20546 df-zring 20618 df-zrh 20651 df-zn 20654 |
This theorem is referenced by: cygth 20718 |
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