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Mirrors > Home > ILE Home > Th. List > znzrh | 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 2194 | . 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 14105 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌)) |
6 | 2, 3, 4 | znadd 14106 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌)) |
7 | 6 | oveqdr 5938 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(+g‘𝑈)𝑦) = (𝑥(+g‘𝑌)𝑦)) |
8 | 2, 3, 4 | znmul 14107 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌)) |
9 | 8 | oveqdr 5938 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(.r‘𝑈)𝑦) = (𝑥(.r‘𝑌)𝑦)) |
10 | 1, 5, 7, 9 | zrhpropd 14091 | 1 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) = (ℤRHom‘𝑌)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 {csn 3618 ‘cfv 5246 (class class class)co 5910 ℕ0cn0 9230 Basecbs 12608 +gcplusg 12685 .rcmulr 12686 /s cqus 12873 ~QG cqg 13228 RSpancrsp 13948 ℤringczring 14056 ℤRHomczrh 14076 ℤ/nℤczn 14078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-mulrcl 7961 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-precex 7972 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 ax-pre-mulgt0 7979 ax-addf 7984 ax-mulf 7985 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-tp 3626 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-ec 6580 df-map 6695 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-reap 8584 df-inn 8973 df-2 9031 df-3 9032 df-4 9033 df-5 9034 df-6 9035 df-7 9036 df-8 9037 df-9 9038 df-n0 9231 df-z 9308 df-dec 9439 df-uz 9583 df-fz 10065 df-cj 10976 df-struct 12610 df-ndx 12611 df-slot 12612 df-base 12614 df-sets 12615 df-iress 12616 df-plusg 12698 df-mulr 12699 df-starv 12700 df-sca 12701 df-vsca 12702 df-ip 12703 df-ple 12705 df-0g 12859 df-iimas 12875 df-qus 12876 df-mgm 12929 df-sgrp 12975 df-mnd 12988 df-mhm 13021 df-grp 13065 df-minusg 13066 df-subg 13229 df-eqg 13231 df-ghm 13300 df-cmn 13345 df-mgp 13401 df-ur 13440 df-ring 13478 df-cring 13479 df-rhm 13632 df-subrg 13699 df-lsp 13867 df-sra 13915 df-rgmod 13916 df-rsp 13950 df-icnfld 14032 df-zring 14057 df-zrh 14079 df-zn 14081 |
This theorem is referenced by: znzrh2 14111 znle2 14117 |
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