![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2734 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) | |
2 | zringbas 21008 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤ = (Base‘ℤring)) |
4 | eqid 2733 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) | |
5 | ovexd 7439 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) ∈ V) | |
6 | zringring 21005 | . . . . 5 ⊢ ℤring ∈ Ring | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤring ∈ Ring) |
8 | 1, 3, 4, 5, 7 | quslem 17485 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→(ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
9 | eqid 2733 | . . . . . 6 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring) | |
10 | znzrhfo.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
11 | eqid 2733 | . . . . . 6 ⊢ (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) | |
12 | 9, 10, 11 | znbas 21083 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (Base‘𝑌)) |
13 | znzrhfo.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
14 | 12, 13 | eqtr4di 2791 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = 𝐵) |
15 | foeq3 6800 | . . . 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 231 | . 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 21085 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
20 | foeq1 6798 | . . 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 257 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4627 ↦ cmpt 5230 –onto→wfo 6538 ‘cfv 6540 (class class class)co 7404 [cec 8697 / cqs 8698 ℕ0cn0 12468 ℤcz 12554 Basecbs 17140 /s cqus 17447 ~QG cqg 18996 Ringcrg 20047 RSpancrsp 20772 ℤringczring 21002 ℤRHomczrh 21033 ℤ/nℤczn 21036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-seq 13963 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-imas 17450 df-qus 17451 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-nsg 18998 df-eqg 18999 df-ghm 19084 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-oppr 20139 df-rnghom 20240 df-subrg 20349 df-lmod 20461 df-lss 20531 df-lsp 20571 df-sra 20773 df-rgmod 20774 df-lidl 20775 df-rsp 20776 df-2idl 20844 df-cnfld 20930 df-zring 21003 df-zrh 21037 df-zn 21040 |
This theorem is referenced by: zncyg 21088 znf1o 21091 zzngim 21092 znfld 21100 znunit 21103 znrrg 21105 cygznlem2a 21107 cygznlem3 21109 dchrelbas4 26726 dchrzrhcl 26728 lgsdchrval 26837 lgsdchr 26838 rpvmasumlem 26970 dchrmusum2 26977 dchrvmasumlem3 26982 dchrisum0ff 26990 dchrisum0flblem1 26991 rpvmasum2 26995 dchrisum0re 26996 dchrisum0lem2a 27000 dirith 27012 znfermltl 32448 |
Copyright terms: Public domain | W3C validator |