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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 2741 | . 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 20742 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌)) |
6 | 2, 3, 4 | znadd 20744 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌)) |
7 | 6 | oveqdr 7299 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(+g‘𝑈)𝑦) = (𝑥(+g‘𝑌)𝑦)) |
8 | 2, 3, 4 | znmul 20746 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌)) |
9 | 8 | oveqdr 7299 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(.r‘𝑈)𝑦) = (𝑥(.r‘𝑌)𝑦)) |
10 | 1, 5, 7, 9 | zrhpropd 20714 | 1 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) = (ℤRHom‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {csn 4567 ‘cfv 6432 (class class class)co 7271 ℕ0cn0 12233 Basecbs 16910 +gcplusg 16960 .rcmulr 16961 /s cqus 17214 ~QG cqg 18749 RSpancrsp 20431 ℤringczring 20668 ℤRHomczrh 20699 ℤ/nℤczn 20702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-grp 18578 df-minusg 18579 df-subg 18750 df-ghm 18830 df-cmn 19386 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-rnghom 19957 df-subrg 20020 df-cnfld 20596 df-zring 20669 df-zrh 20703 df-zn 20706 |
This theorem is referenced by: znzrh2 20751 znle2 20759 |
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