![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > znzrh | Structured version Visualization version GIF version |
Description: The ℤ ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
Ref | Expression |
---|---|
znval2.s | ⊢ 𝑆 = (RSpan‘ℤring) |
znval2.u | ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) |
znval2.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
Ref | Expression |
---|---|
znzrh | ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) = (ℤRHom‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2728 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑈)) | |
2 | znval2.s | . . 3 ⊢ 𝑆 = (RSpan‘ℤring) | |
3 | znval2.u | . . 3 ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) | |
4 | znval2.y | . . 3 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
5 | 2, 3, 4 | znbas2 21463 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌)) |
6 | 2, 3, 4 | znadd 21465 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌)) |
7 | 6 | oveqdr 7442 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(+g‘𝑈)𝑦) = (𝑥(+g‘𝑌)𝑦)) |
8 | 2, 3, 4 | znmul 21467 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌)) |
9 | 8 | oveqdr 7442 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(.r‘𝑈)𝑦) = (𝑥(.r‘𝑌)𝑦)) |
10 | 1, 5, 7, 9 | zrhpropd 21433 | 1 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) = (ℤRHom‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4624 ‘cfv 6542 (class class class)co 7414 ℕ0cn0 12496 Basecbs 17173 +gcplusg 17226 .rcmulr 17227 /s cqus 17480 ~QG cqg 19070 RSpancrsp 21096 ℤringczring 21365 ℤRHomczrh 21418 ℤ/nℤczn 21421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-addf 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-grp 18886 df-minusg 18887 df-subg 19071 df-ghm 19161 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-rhm 20404 df-subrng 20476 df-subrg 20501 df-cnfld 21273 df-zring 21366 df-zrh 21422 df-zn 21425 |
This theorem is referenced by: znzrh2 21472 znle2 21480 |
Copyright terms: Public domain | W3C validator |