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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 2778 | . 2 ⊢ 𝐿 = 𝐿 | |
2 | zrhval.l | . . 3 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
3 | 2 | zrhrhmb 20255 | . 2 ⊢ (𝑅 ∈ Ring → (𝐿 ∈ (ℤring RingHom 𝑅) ↔ 𝐿 = 𝐿)) |
4 | 1, 3 | mpbiri 250 | 1 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 Ringcrg 18934 RingHom crh 19101 ℤringzring 20214 ℤRHomczrh 20244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-seq 13120 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cmn 18581 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-rnghom 19104 df-subrg 19170 df-cnfld 20143 df-zring 20215 df-zrh 20248 |
This theorem is referenced by: zrh1 20257 zrh0 20258 chrrhm 20275 domnchr 20276 zndvds0 20294 znf1o 20295 zzngim 20296 znfld 20304 znidomb 20305 znunit 20307 znrrg 20309 cygznlem3 20313 zrhpsgnmhm 20325 zrhpsgnodpm 20333 dchrzrhmul 25423 lgsqrlem1 25523 lgsqrlem2 25524 lgsqrlem3 25525 lgsdchr 25532 lgseisenlem3 25554 lgseisenlem4 25555 dchrisum0flblem1 25649 mdetpmtr1 30487 mdetpmtr12 30489 mdetlap 30496 zrhf1ker 30617 zrhunitpreima 30620 elzrhunit 30621 qqhval2lem 30623 qqhf 30628 qqhghm 30630 qqhrhm 30631 qqhnm 30632 |
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