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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 2824 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) | |
2 | zringbas 20625 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤ = (Base‘ℤring)) |
4 | eqid 2823 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) | |
5 | ovexd 7193 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) ∈ V) | |
6 | zringring 20622 | . . . . 5 ⊢ ℤring ∈ Ring | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ℤring ∈ Ring) |
8 | 1, 3, 4, 5, 7 | quslem 16818 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))):ℤ–onto→(ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
9 | eqid 2823 | . . . . . 6 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring) | |
10 | znzrhfo.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
11 | eqid 2823 | . . . . . 6 ⊢ (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})) | |
12 | 9, 10, 11 | znbas 20692 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (Base‘𝑌)) |
13 | znzrhfo.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
14 | 12, 13 | syl6eqr 2876 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℤ / (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = 𝐵) |
15 | foeq3 6590 | . . . 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 234 | . 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 20694 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) |
20 | foeq1 6588 | . . 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 259 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 ↦ cmpt 5148 –onto→wfo 6355 ‘cfv 6357 (class class class)co 7158 [cec 8289 / cqs 8290 ℕ0cn0 11900 ℤcz 11984 Basecbs 16485 /s cqus 16780 ~QG cqg 18277 Ringcrg 19299 RSpancrsp 19945 ℤringzring 20619 ℤRHomczrh 20649 ℤ/nℤczn 20652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-seq 13373 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-nsg 18279 df-eqg 18280 df-ghm 18358 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-2idl 20007 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-zn 20656 |
This theorem is referenced by: zncyg 20697 znf1o 20700 zzngim 20701 znfld 20709 znunit 20712 znrrg 20714 cygznlem2a 20716 cygznlem3 20718 dchrelbas4 25821 dchrzrhcl 25823 lgsdchrval 25932 lgsdchr 25933 rpvmasumlem 26065 dchrmusum2 26072 dchrvmasumlem3 26077 dchrisum0ff 26085 dchrisum0flblem1 26086 rpvmasum2 26090 dchrisum0re 26091 dchrisum0lem2a 26095 dirith 26107 |
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