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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 21336 | . . . . 5 ⊢ ℤring ∈ Ring | |
3 | nn0z 12587 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
4 | znzrh2.s | . . . . . . 7 ⊢ 𝑆 = (RSpan‘ℤring) | |
5 | 4 | znlidl 21424 | . . . . . 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 7416 | . . . . . 6 ⊢ (ℤring /s ∼ ) = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) |
9 | zringcrng 21335 | . . . . . . 7 ⊢ ℤring ∈ CRing | |
10 | eqid 2726 | . . . . . . . 8 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
11 | 10 | crng2idl 21136 | . . . . . . 7 ⊢ (ℤring ∈ CRing → (LIdeal‘ℤring) = (2Ideal‘ℤring)) |
12 | 9, 11 | ax-mp 5 | . . . . . 6 ⊢ (LIdeal‘ℤring) = (2Ideal‘ℤring) |
13 | zringbas 21340 | . . . . . 6 ⊢ ℤ = (Base‘ℤring) | |
14 | eceq2 8745 | . . . . . . . 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 5246 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (𝑥 ∈ ℤ ↦ [𝑥](ℤring ~QG (𝑆‘{𝑁}))) |
17 | 8, 12, 13, 16 | qusrhm 21133 | . . . . 5 ⊢ ((ℤring ∈ Ring ∧ (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring)) → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤring RingHom (ℤring /s ∼ ))) |
18 | 2, 6, 17 | sylancr 586 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤring RingHom (ℤring /s ∼ ))) |
19 | 4, 8 | zncrng2 21425 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (ℤring /s ∼ ) ∈ CRing) |
20 | crngring 20150 | . . . . 5 ⊢ ((ℤring /s ∼ ) ∈ CRing → (ℤring /s ∼ ) ∈ Ring) | |
21 | eqid 2726 | . . . . . 6 ⊢ (ℤRHom‘(ℤring /s ∼ )) = (ℤRHom‘(ℤring /s ∼ )) | |
22 | 21 | zrhrhmb 21397 | . . . . 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 231 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (ℤRHom‘(ℤring /s ∼ ))) |
25 | znzrh2.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
26 | 4, 8, 25 | znzrh 21437 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘(ℤring /s ∼ )) = (ℤRHom‘𝑌)) |
27 | 24, 26 | eqtr2d 2767 | . 2 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌) = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) |
28 | 1, 27 | eqtrid 2778 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {csn 4623 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 [cec 8703 ℕ0cn0 12476 ℤcz 12562 /s cqus 17460 ~QG cqg 19049 Ringcrg 20138 CRingccrg 20139 RingHom crh 20371 LIdealclidl 21065 RSpancrsp 21066 2Idealc2idl 21106 ℤringczring 21333 ℤRHomczrh 21386 ℤ/nℤczn 21389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-seq 13973 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-0g 17396 df-imas 17463 df-qus 17464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-nsg 19051 df-eqg 19052 df-ghm 19139 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-lsp 20819 df-sra 21021 df-rgmod 21022 df-lidl 21067 df-rsp 21068 df-2idl 21107 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-zn 21393 |
This theorem is referenced by: znzrhval 21441 znzrhfo 21442 |
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