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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 12539 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | zzngim.y | . . . . 5 ⊢ 𝑌 = (ℤ/nℤ‘0) | |
3 | 2 | zncrng 21581 | . . . 4 ⊢ (0 ∈ ℕ0 → 𝑌 ∈ CRing) |
4 | crngring 20263 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ 𝑌 ∈ Ring |
6 | zzngim.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
7 | 6 | zrhrhm 21540 | . . 3 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌)) |
8 | rhmghm 20501 | . . 3 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌)) | |
9 | 5, 7, 8 | mp2b 10 | . 2 ⊢ 𝐿 ∈ (ℤring GrpHom 𝑌) |
10 | eqid 2735 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
11 | 2, 10, 6 | znzrhfo 21584 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌)) |
12 | 1, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐿:ℤ–onto→(Base‘𝑌) |
13 | fofn 6823 | . . . . . 6 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿 Fn ℤ) | |
14 | fnresdm 6688 | . . . . . 6 ⊢ (𝐿 Fn ℤ → (𝐿 ↾ ℤ) = 𝐿) | |
15 | 12, 13, 14 | mp2b 10 | . . . . 5 ⊢ (𝐿 ↾ ℤ) = 𝐿 |
16 | 6 | reseq1i 5996 | . . . . 5 ⊢ (𝐿 ↾ ℤ) = ((ℤRHom‘𝑌) ↾ ℤ) |
17 | 15, 16 | eqtr3i 2765 | . . . 4 ⊢ 𝐿 = ((ℤRHom‘𝑌) ↾ ℤ) |
18 | eqid 2735 | . . . . . 6 ⊢ 0 = 0 | |
19 | 18 | iftruei 4538 | . . . . 5 ⊢ if(0 = 0, ℤ, (0..^0)) = ℤ |
20 | 19 | eqcomi 2744 | . . . 4 ⊢ ℤ = if(0 = 0, ℤ, (0..^0)) |
21 | 2, 10, 17, 20 | znf1o 21588 | . . 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 21482 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
24 | 23, 10 | isgim 19293 | . 2 ⊢ (𝐿 ∈ (ℤring GrpIso 𝑌) ↔ (𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝐿:ℤ–1-1-onto→(Base‘𝑌))) |
25 | 9, 22, 24 | mpbir2an 711 | 1 ⊢ 𝐿 ∈ (ℤring GrpIso 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ifcif 4531 ↾ cres 5691 Fn wfn 6558 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ℕ0cn0 12524 ℤcz 12611 ..^cfzo 13691 Basecbs 17245 GrpHom cghm 19243 GrpIso cgim 19288 Ringcrg 20251 CRingccrg 20252 RingHom crh 20486 ℤringczring 21475 ℤRHomczrh 21528 ℤ/nℤczn 21531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-dvds 16288 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-gim 19290 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zn 21535 |
This theorem is referenced by: (None) |
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