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Mirrors > Home > MPE Home > Th. List > zrhrhm | Structured version Visualization version GIF version |
Description: The ℤRHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
Ref | Expression |
---|---|
zrhval.l | ⊢ 𝐿 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
zrhrhm | ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . 2 ⊢ 𝐿 = 𝐿 | |
2 | zrhval.l | . . 3 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
3 | 2 | zrhrhmb 21500 | . 2 ⊢ (𝑅 ∈ Ring → (𝐿 ∈ (ℤring RingHom 𝑅) ↔ 𝐿 = 𝐿)) |
4 | 1, 3 | mpbiri 257 | 1 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Ringcrg 20216 RingHom crh 20451 ℤringczring 21436 ℤRHomczrh 21489 |
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 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-addf 11237 ax-mulf 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-seq 14022 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-grp 18931 df-minusg 18932 df-mulg 19062 df-subg 19117 df-ghm 19207 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 df-rhm 20454 df-subrng 20528 df-subrg 20553 df-cnfld 21344 df-zring 21437 df-zrh 21493 |
This theorem is referenced by: zrh1 21502 zrh0 21503 fermltlchr 21523 chrrhm 21525 domnchr 21526 zndvds0 21548 znf1o 21549 zzngim 21550 znfld 21558 znidomb 21559 znunit 21561 znrrg 21563 cygznlem3 21567 zrhpsgnmhm 21580 zrhpsgnodpm 21588 ply1fermltlchr 22303 dchrzrhmul 27275 lgsqrlem1 27375 lgsqrlem2 27376 lgsqrlem3 27377 lgsdchr 27384 lgseisenlem3 27406 lgseisenlem4 27407 dchrisum0flblem1 27537 znfermltl 33241 elrspunidl 33303 mdetpmtr1 33638 mdetpmtr12 33640 mdetlap 33647 zrhf1ker 33790 zrhunitpreima 33793 elzrhunit 33794 qqhval2lem 33796 qqhf 33801 qqhghm 33803 qqhrhm 33804 qqhnm 33805 zndvdchrrhm 41669 aks6d1c1p2 41807 aks6d1c1p3 41808 aks6d1c1 41814 hashscontpowcl 41818 hashscontpow 41820 aks6d1c4 41822 aks6d1c2 41828 aks6d1c5lem0 41833 aks6d1c5lem1 41834 aks6d1c5lem3 41835 aks6d1c5lem2 41836 aks6d1c5 41837 aks6d1c6lem1 41868 aks6d1c6lem3 41870 aks6d1c6lem5 41875 aks6d1c7lem1 41878 aks5lem3a 41887 aks5lem5a 41889 |
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