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Mirrors > Home > MPE Home > Th. List > znzrhfo | Structured version Visualization version GIF version |
Description: The ℤ ring homomorphism is a surjection onto ℤ / 𝑛ℤ. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
znzrhfo.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
znzrhfo.b | ⊢ 𝐵 = (Base‘𝑌) |
znzrhfo.2 | ⊢ 𝐿 = (ℤRHom‘𝑌) |
Ref | Expression |
---|---|
znzrhfo | ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2826 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) | |
2 | zringbas 20184 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤ = (Base‘ℤring)) |
4 | eqid 2825 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) | |
5 | ovexd 6939 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) ∈ V) | |
6 | zringring 20181 | . . . . 5 ⊢ ℤring ∈ Ring | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤring ∈ Ring) |
8 | 1, 3, 4, 5, 7 | quslem 16556 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→(ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
9 | eqid 2825 | . . . . . 6 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring) | |
10 | znzrhfo.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
11 | eqid 2825 | . . . . . 6 ⊢ (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) | |
12 | 9, 10, 11 | znbas 20251 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (Base‘𝑌)) |
13 | znzrhfo.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
14 | 12, 13 | syl6eqr 2879 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = 𝐵) |
15 | foeq3 6351 | . . . 4 ⊢ ((ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = 𝐵 → ((𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→(ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) ↔ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→𝐵)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→(ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) ↔ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→𝐵)) |
17 | 8, 16 | mpbid 224 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→𝐵) |
18 | znzrhfo.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
19 | 9, 11, 10, 18 | znzrh2 20253 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
20 | foeq1 6349 | . . 3 ⊢ (𝐿 = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) → (𝐿:ℤ–onto→𝐵 ↔ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→𝐵)) | |
21 | 19, 20 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐿:ℤ–onto→𝐵 ↔ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→𝐵)) |
22 | 17, 21 | mpbird 249 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∈ wcel 2166 Vcvv 3414 {csn 4397 ↦ cmpt 4952 –onto→wfo 6121 ‘cfv 6123 (class class class)co 6905 [cec 8007 / cqs 8008 ℕ0cn0 11618 ℤcz 11704 Basecbs 16222 /s cqus 16518 ~QG cqg 17941 Ringcrg 18901 RSpancrsp 19532 ℤringzring 20178 ℤRHomczrh 20208 ℤ/nℤczn 20211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-ec 8011 df-qs 8015 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-seq 13096 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-0g 16455 df-imas 16521 df-qus 16522 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-minusg 17780 df-sbg 17781 df-mulg 17895 df-subg 17942 df-nsg 17943 df-eqg 17944 df-ghm 18009 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-cring 18904 df-oppr 18977 df-rnghom 19071 df-subrg 19134 df-lmod 19221 df-lss 19289 df-lsp 19331 df-sra 19533 df-rgmod 19534 df-lidl 19535 df-rsp 19536 df-2idl 19593 df-cnfld 20107 df-zring 20179 df-zrh 20212 df-zn 20215 |
This theorem is referenced by: zncyg 20256 znf1o 20259 zzngim 20260 znfld 20268 znunit 20271 znrrg 20273 cygznlem2a 20275 cygznlem3 20277 dchrelbas4 25381 dchrzrhcl 25383 lgsdchrval 25492 lgsdchr 25493 rpvmasumlem 25589 dchrmusum2 25596 dchrvmasumlem3 25601 dchrisum0ff 25609 dchrisum0flblem1 25610 rpvmasum2 25614 dchrisum0re 25615 dchrisum0lem2a 25619 dirith 25631 |
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