Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20693 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
3 | crngring 19310 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Ring) |
5 | ringgrp 19304 | . . 3 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Grp) |
7 | eqid 2823 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
8 | eqid 2823 | . . . . 5 ⊢ (1r‘𝑌) = (1r‘𝑌) | |
9 | 7, 8 | ringidcl 19320 | . . . 4 ⊢ (𝑌 ∈ Ring → (1r‘𝑌) ∈ (Base‘𝑌)) |
10 | 4, 9 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1r‘𝑌) ∈ (Base‘𝑌)) |
11 | eqid 2823 | . . . . . . 7 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌) | |
12 | eqid 2823 | . . . . . . 7 ⊢ (.g‘𝑌) = (.g‘𝑌) | |
13 | 11, 12, 8 | zrhval2 20658 | . . . . . 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 5810 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ran (ℤRHom‘𝑌) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
16 | 1, 7, 11 | znzrhfo 20696 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌)) |
17 | forn 6595 | . . . . 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 2860 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌)) |
20 | oveq2 7166 | . . . . . . 7 ⊢ (𝑥 = (1r‘𝑌) → (𝑛(.g‘𝑌)𝑥) = (𝑛(.g‘𝑌)(1r‘𝑌))) | |
21 | 20 | mpteq2dv 5164 | . . . . . 6 ⊢ (𝑥 = (1r‘𝑌) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
22 | 21 | rneqd 5810 | . . . . 5 ⊢ (𝑥 = (1r‘𝑌) → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
23 | 22 | eqeq1d 2825 | . . . 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 3625 | . . 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 19000 | . 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 3141 ↦ cmpt 5148 ran crn 5558 –onto→wfo 6355 ‘cfv 6357 (class class class)co 7158 ℕ0cn0 11900 ℤcz 11984 Basecbs 16485 Grpcgrp 18105 .gcmg 18226 CycGrpccyg 18998 1rcur 19253 Ringcrg 19299 CRingccrg 19300 ℤRHomczrh 20649 ℤ/nℤczn 20652 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-seq 13373 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-nsg 18279 df-eqg 18280 df-ghm 18358 df-cmn 18910 df-abl 18911 df-cyg 18999 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-2idl 20007 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-zn 20656 |
This theorem is referenced by: cygth 20720 |
Copyright terms: Public domain | W3C validator |