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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 2730 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) | |
| 2 | zringbas 21395 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤ = (Base‘ℤring)) |
| 4 | eqid 2729 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) | |
| 5 | ovexd 7404 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) ∈ V) | |
| 6 | zringring 21391 | . . . . 5 ⊢ ℤring ∈ Ring | |
| 7 | 6 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤring ∈ Ring) |
| 8 | 1, 3, 4, 5, 7 | quslem 17482 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→(ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
| 9 | eqid 2729 | . . . . . 6 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring) | |
| 10 | znzrhfo.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
| 11 | eqid 2729 | . . . . . 6 ⊢ (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) | |
| 12 | 9, 10, 11 | znbas 21485 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (Base‘𝑌)) |
| 13 | znzrhfo.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
| 14 | 12, 13 | eqtr4di 2782 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = 𝐵) |
| 15 | foeq3 6752 | . . . 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 232 | . 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 21487 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
| 20 | foeq1 6750 | . . 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 206 = wceq 1540 ∈ wcel 2109 Vcvv 3444 {csn 4585 ↦ cmpt 5183 –onto→wfo 6497 ‘cfv 6499 (class class class)co 7369 [cec 8646 / cqs 8647 ℕ0cn0 12418 ℤcz 12505 Basecbs 17155 /s cqus 17444 ~QG cqg 19036 Ringcrg 20153 RSpancrsp 21149 ℤringczring 21388 ℤRHomczrh 21441 ℤ/nℤczn 21444 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-addf 11123 ax-mulf 11124 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-seq 13943 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17380 df-imas 17447 df-qus 17448 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-nsg 19038 df-eqg 19039 df-ghm 19127 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-rhm 20392 df-subrng 20466 df-subrg 20490 df-lmod 20800 df-lss 20870 df-lsp 20910 df-sra 21112 df-rgmod 21113 df-lidl 21150 df-rsp 21151 df-2idl 21192 df-cnfld 21297 df-zring 21389 df-zrh 21445 df-zn 21448 |
| This theorem is referenced by: zncyg 21490 znf1o 21493 zzngim 21494 znfld 21502 znunit 21505 znrrg 21507 cygznlem2a 21509 cygznlem3 21511 dchrelbas4 27187 dchrzrhcl 27189 lgsdchrval 27298 lgsdchr 27299 rpvmasumlem 27431 dchrmusum2 27438 dchrvmasumlem3 27443 dchrisum0ff 27451 dchrisum0flblem1 27452 rpvmasum2 27456 dchrisum0re 27457 dchrisum0lem2a 27461 dirith 27473 znfermltl 33330 |
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