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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 20236 | . . . 4 ⊢ (0 ∈ ℕ0 → 𝑌 ∈ CRing) |
4 | crngring 19302 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ 𝑌 ∈ Ring |
6 | zzngim.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
7 | 6 | zrhrhm 20205 | . . 3 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
8 | rhmghm 19473 | . . 3 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌)) | |
9 | 5, 7, 8 | mp2b 10 | . 2 ⊢ 𝐿 ∈ (ℤring GrpHom 𝑌) |
10 | eqid 2798 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
11 | 2, 10, 6 | znzrhfo 20239 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌)) |
12 | 1, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐿:ℤ–onto→(Base‘𝑌) |
13 | fofn 6567 | . . . . . 6 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿 Fn ℤ) | |
14 | fnresdm 6438 | . . . . . 6 ⊢ (𝐿 Fn ℤ → (𝐿 ↾ ℤ) = 𝐿) | |
15 | 12, 13, 14 | mp2b 10 | . . . . 5 ⊢ (𝐿 ↾ ℤ) = 𝐿 |
16 | 6 | reseq1i 5814 | . . . . 5 ⊢ (𝐿 ↾ ℤ) = ((ℤRHom‘𝑌) ↾ ℤ) |
17 | 15, 16 | eqtr3i 2823 | . . . 4 ⊢ 𝐿 = ((ℤRHom‘𝑌) ↾ ℤ) |
18 | eqid 2798 | . . . . . 6 ⊢ 0 = 0 | |
19 | 18 | iftruei 4432 | . . . . 5 ⊢ if(0 = 0, ℤ, (0..^0)) = ℤ |
20 | 19 | eqcomi 2807 | . . . 4 ⊢ ℤ = if(0 = 0, ℤ, (0..^0)) |
21 | 2, 10, 17, 20 | znf1o 20243 | . . 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 20169 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
24 | 23, 10 | isgim 18394 | . 2 ⊢ (𝐿 ∈ (ℤring GrpIso 𝑌) ↔ (𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝐿:ℤ–1-1-onto→(Base‘𝑌))) |
25 | 9, 22, 24 | mpbir2an 710 | 1 ⊢ 𝐿 ∈ (ℤring GrpIso 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ifcif 4425 ↾ cres 5521 Fn wfn 6319 –onto→wfo 6322 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℕ0cn0 11885 ℤcz 11969 ..^cfzo 13028 Basecbs 16475 GrpHom cghm 18347 GrpIso cgim 18389 Ringcrg 19290 CRingccrg 19291 RingHom crh 19460 ℤringzring 20163 ℤRHomczrh 20193 ℤ/nℤczn 20196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-dvds 15600 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-imas 16773 df-qus 16774 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-nsg 18269 df-eqg 18270 df-ghm 18348 df-gim 18391 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-rnghom 19463 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 df-2idl 19998 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-zn 20200 |
This theorem is referenced by: (None) |
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