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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 11184 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | zzngim.y | . . . . 5 ⊢ 𝑌 = (ℤ/nℤ‘0) | |
3 | 2 | zncrng 19712 | . . . 4 ⊢ (0 ∈ ℕ0 → 𝑌 ∈ CRing) |
4 | crngring 18381 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ 𝑌 ∈ Ring |
6 | zzngim.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
7 | 6 | zrhrhm 19679 | . . 3 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
8 | rhmghm 18548 | . . 3 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌)) | |
9 | 5, 7, 8 | mp2b 10 | . 2 ⊢ 𝐿 ∈ (ℤring GrpHom 𝑌) |
10 | eqid 2610 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
11 | 2, 10, 6 | znzrhfo 19715 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌)) |
12 | 1, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐿:ℤ–onto→(Base‘𝑌) |
13 | fofn 6030 | . . . . . 6 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿 Fn ℤ) | |
14 | fnresdm 5914 | . . . . . 6 ⊢ (𝐿 Fn ℤ → (𝐿 ↾ ℤ) = 𝐿) | |
15 | 12, 13, 14 | mp2b 10 | . . . . 5 ⊢ (𝐿 ↾ ℤ) = 𝐿 |
16 | 6 | reseq1i 5313 | . . . . 5 ⊢ (𝐿 ↾ ℤ) = ((ℤRHom‘𝑌) ↾ ℤ) |
17 | 15, 16 | eqtr3i 2634 | . . . 4 ⊢ 𝐿 = ((ℤRHom‘𝑌) ↾ ℤ) |
18 | eqid 2610 | . . . . . 6 ⊢ 0 = 0 | |
19 | 18 | iftruei 4043 | . . . . 5 ⊢ if(0 = 0, ℤ, (0..^0)) = ℤ |
20 | 19 | eqcomi 2619 | . . . 4 ⊢ ℤ = if(0 = 0, ℤ, (0..^0)) |
21 | 2, 10, 17, 20 | znf1o 19719 | . . 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 19643 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
24 | 23, 10 | isgim 17527 | . 2 ⊢ (𝐿 ∈ (ℤring GrpIso 𝑌) ↔ (𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝐿:ℤ–1-1-onto→(Base‘𝑌))) |
25 | 9, 22, 24 | mpbir2an 957 | 1 ⊢ 𝐿 ∈ (ℤring GrpIso 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ifcif 4036 ↾ cres 5040 Fn wfn 5799 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℕ0cn0 11169 ℤcz 11254 ..^cfzo 12334 Basecbs 15695 GrpHom cghm 17480 GrpIso cgim 17522 Ringcrg 18370 CRingccrg 18371 RingHom crh 18535 ℤringzring 19637 ℤRHomczrh 19667 ℤ/nℤczn 19670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-dvds 14822 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-gim 17524 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 |
This theorem is referenced by: (None) |
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