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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 2799 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) | |
2 | zringbas 20169 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤ = (Base‘ℤring)) |
4 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) | |
5 | ovexd 7170 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) ∈ V) | |
6 | zringring 20166 | . . . . 5 ⊢ ℤring ∈ Ring | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤring ∈ Ring) |
8 | 1, 3, 4, 5, 7 | quslem 16808 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→(ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
9 | eqid 2798 | . . . . . 6 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring) | |
10 | znzrhfo.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
11 | eqid 2798 | . . . . . 6 ⊢ (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) | |
12 | 9, 10, 11 | znbas 20235 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (Base‘𝑌)) |
13 | znzrhfo.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
14 | 12, 13 | eqtr4di 2851 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = 𝐵) |
15 | foeq3 6563 | . . . 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 235 | . 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 20237 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
20 | foeq1 6561 | . . 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 260 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 ↦ cmpt 5110 –onto→wfo 6322 ‘cfv 6324 (class class class)co 7135 [cec 8270 / cqs 8271 ℕ0cn0 11885 ℤcz 11969 Basecbs 16475 /s cqus 16770 ~QG cqg 18267 Ringcrg 19290 RSpancrsp 19936 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-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-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-fz 12886 df-seq 13365 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-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 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: zncyg 20240 znf1o 20243 zzngim 20244 znfld 20252 znunit 20255 znrrg 20257 cygznlem2a 20259 cygznlem3 20261 dchrelbas4 25827 dchrzrhcl 25829 lgsdchrval 25938 lgsdchr 25939 rpvmasumlem 26071 dchrmusum2 26078 dchrvmasumlem3 26083 dchrisum0ff 26091 dchrisum0flblem1 26092 rpvmasum2 26096 dchrisum0re 26097 dchrisum0lem2a 26101 dirith 26113 |
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