Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > znzrh2 | Structured version Visualization version GIF version |
Description: The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
Ref | Expression |
---|---|
znzrh2.s | ⊢ 𝑆 = (RSpan‘ℤring) |
znzrh2.r | ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) |
znzrh2.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
znzrh2.2 | ⊢ 𝐿 = (ℤRHom‘𝑌) |
Ref | Expression |
---|---|
znzrh2 | ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znzrh2.2 | . 2 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
2 | zringring 20548 | . . . . 5 ⊢ ℤring ∈ Ring | |
3 | nn0z 11993 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
4 | znzrh2.s | . . . . . . 7 ⊢ 𝑆 = (RSpan‘ℤring) | |
5 | 4 | znlidl 20608 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring)) |
6 | 3, 5 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring)) |
7 | znzrh2.r | . . . . . . 7 ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) | |
8 | 7 | oveq2i 7156 | . . . . . 6 ⊢ (ℤring /s ∼ ) = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) |
9 | zringcrng 20547 | . . . . . . 7 ⊢ ℤring ∈ CRing | |
10 | eqid 2818 | . . . . . . . 8 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
11 | 10 | crng2idl 19940 | . . . . . . 7 ⊢ (ℤring ∈ CRing → (LIdeal‘ℤring) = (2Ideal‘ℤring)) |
12 | 9, 11 | ax-mp 5 | . . . . . 6 ⊢ (LIdeal‘ℤring) = (2Ideal‘ℤring) |
13 | zringbas 20551 | . . . . . 6 ⊢ ℤ = (Base‘ℤring) | |
14 | eceq2 8318 | . . . . . . . 8 ⊢ ( ∼ = (ℤring ~QG (𝑆‘{𝑁})) → [𝑥] ∼ = [𝑥](ℤring ~QG (𝑆‘{𝑁}))) | |
15 | 7, 14 | ax-mp 5 | . . . . . . 7 ⊢ [𝑥] ∼ = [𝑥](ℤring ~QG (𝑆‘{𝑁})) |
16 | 15 | mpteq2i 5149 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG (𝑆‘{𝑁}))) |
17 | 8, 12, 13, 16 | qusrhm 19938 | . . . . 5 ⊢ ((ℤring ∈ Ring ∧ (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring)) → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤring RingHom (ℤring /s ∼ ))) |
18 | 2, 6, 17 | sylancr 587 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤring RingHom (ℤring /s ∼ ))) |
19 | 4, 8 | zncrng2 20609 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (ℤring /s ∼ ) ∈ CRing) |
20 | crngring 19237 | . . . . 5 ⊢ ((ℤring /s ∼ ) ∈ CRing → (ℤring /s ∼ ) ∈ Ring) | |
21 | eqid 2818 | . . . . . 6 ⊢ (ℤRHom‘(ℤring /s ∼ )) = (ℤRHom‘(ℤring /s ∼ )) | |
22 | 21 | zrhrhmb 20586 | . . . . 5 ⊢ ((ℤring /s ∼ ) ∈ Ring → ((𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤring RingHom (ℤring /s ∼ )) ↔ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (ℤRHom‘(ℤring /s ∼ )))) |
23 | 3, 19, 20, 22 | 4syl 19 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤring RingHom (ℤring /s ∼ )) ↔ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (ℤRHom‘(ℤring /s ∼ )))) |
24 | 18, 23 | mpbid 233 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (ℤRHom‘(ℤring /s ∼ ))) |
25 | znzrh2.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
26 | 4, 8, 25 | znzrh 20617 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘(ℤring /s ∼ )) = (ℤRHom‘𝑌)) |
27 | 24, 26 | eqtr2d 2854 | . 2 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌) = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) |
28 | 1, 27 | syl5eq 2865 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {csn 4557 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 [cec 8276 ℕ0cn0 11885 ℤcz 11969 /s cqus 16766 ~QG cqg 18213 Ringcrg 19226 CRingccrg 19227 RingHom crh 19393 LIdealclidl 19871 RSpancrsp 19872 2Idealc2idl 19932 ℤringzring 20545 ℤRHomczrh 20575 ℤ/nℤczn 20578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 12881 df-seq 13358 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-imas 16769 df-qus 16770 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-nsg 18215 df-eqg 18216 df-ghm 18294 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-sra 19873 df-rgmod 19874 df-lidl 19875 df-rsp 19876 df-2idl 19933 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-zn 20582 |
This theorem is referenced by: znzrhval 20621 znzrhfo 20622 |
Copyright terms: Public domain | W3C validator |