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Mirrors > Home > MPE Home > Th. List > zzngim | Structured version Visualization version GIF version |
Description: The ℤ ring homomorphism is an isomorphism for 𝑁 = 0. (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 13-Jun-2019.) |
Ref | Expression |
---|---|
zzngim.y | ⊢ 𝑌 = (ℤ/nℤ‘0) |
zzngim.2 | ⊢ 𝐿 = (ℤRHom‘𝑌) |
Ref | Expression |
---|---|
zzngim | ⊢ 𝐿 ∈ (ℤring GrpIso 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11900 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | zzngim.y | . . . . 5 ⊢ 𝑌 = (ℤ/nℤ‘0) | |
3 | 2 | zncrng 20619 | . . . 4 ⊢ (0 ∈ ℕ0 → 𝑌 ∈ CRing) |
4 | crngring 19237 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ 𝑌 ∈ Ring |
6 | zzngim.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
7 | 6 | zrhrhm 20587 | . . 3 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
8 | rhmghm 19406 | . . 3 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌)) | |
9 | 5, 7, 8 | mp2b 10 | . 2 ⊢ 𝐿 ∈ (ℤring GrpHom 𝑌) |
10 | eqid 2818 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
11 | 2, 10, 6 | znzrhfo 20622 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌)) |
12 | 1, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐿:ℤ–onto→(Base‘𝑌) |
13 | fofn 6585 | . . . . . 6 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿 Fn ℤ) | |
14 | fnresdm 6459 | . . . . . 6 ⊢ (𝐿 Fn ℤ → (𝐿 ↾ ℤ) = 𝐿) | |
15 | 12, 13, 14 | mp2b 10 | . . . . 5 ⊢ (𝐿 ↾ ℤ) = 𝐿 |
16 | 6 | reseq1i 5842 | . . . . 5 ⊢ (𝐿 ↾ ℤ) = ((ℤRHom‘𝑌) ↾ ℤ) |
17 | 15, 16 | eqtr3i 2843 | . . . 4 ⊢ 𝐿 = ((ℤRHom‘𝑌) ↾ ℤ) |
18 | eqid 2818 | . . . . . 6 ⊢ 0 = 0 | |
19 | 18 | iftruei 4470 | . . . . 5 ⊢ if(0 = 0, ℤ, (0..^0)) = ℤ |
20 | 19 | eqcomi 2827 | . . . 4 ⊢ ℤ = if(0 = 0, ℤ, (0..^0)) |
21 | 2, 10, 17, 20 | znf1o 20626 | . . 3 ⊢ (0 ∈ ℕ0 → 𝐿:ℤ–1-1-onto→(Base‘𝑌)) |
22 | 1, 21 | ax-mp 5 | . 2 ⊢ 𝐿:ℤ–1-1-onto→(Base‘𝑌) |
23 | zringbas 20551 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
24 | 23, 10 | isgim 18340 | . 2 ⊢ (𝐿 ∈ (ℤring GrpIso 𝑌) ↔ (𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝐿:ℤ–1-1-onto→(Base‘𝑌))) |
25 | 9, 22, 24 | mpbir2an 707 | 1 ⊢ 𝐿 ∈ (ℤring GrpIso 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ifcif 4463 ↾ cres 5550 Fn wfn 6343 –onto→wfo 6346 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ℕ0cn0 11885 ℤcz 11969 ..^cfzo 13021 Basecbs 16471 GrpHom cghm 18293 GrpIso cgim 18335 Ringcrg 19226 CRingccrg 19227 RingHom crh 19393 ℤringzring 20545 ℤRHomczrh 20575 ℤ/nℤczn 20578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-dvds 15596 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-imas 16769 df-qus 16770 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-nsg 18215 df-eqg 18216 df-ghm 18294 df-gim 18337 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-sra 19873 df-rgmod 19874 df-lidl 19875 df-rsp 19876 df-2idl 19933 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-zn 20582 |
This theorem is referenced by: (None) |
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