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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 2728 | . 2 ⊢ 𝐿 = 𝐿 | |
| 2 | zrhval.l | . . 3 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
| 3 | 2 | zrhrhmb 21436 | . 2 ⊢ (𝑅 ∈ Ring → (𝐿 ∈ (ℤring RingHom 𝑅) ↔ 𝐿 = 𝐿)) |
| 4 | 1, 3 | mpbiri 258 | 1 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Ringcrg 20173 RingHom crh 20408 ℤringczring 21372 ℤRHomczrh 21425 |
| 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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 ax-mulf 11219 |
| 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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-seq 14000 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-grp 18893 df-minusg 18894 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-cnfld 21280 df-zring 21373 df-zrh 21429 |
| This theorem is referenced by: zrh1 21438 zrh0 21439 fermltlchr 21459 chrrhm 21461 domnchr 21462 zndvds0 21484 znf1o 21485 zzngim 21486 znfld 21494 znidomb 21495 znunit 21497 znrrg 21499 cygznlem3 21503 zrhpsgnmhm 21516 zrhpsgnodpm 21524 ply1fermltlchr 22231 dchrzrhmul 27192 lgsqrlem1 27292 lgsqrlem2 27293 lgsqrlem3 27294 lgsdchr 27301 lgseisenlem3 27323 lgseisenlem4 27324 dchrisum0flblem1 27454 znfermltl 33091 elrspunidl 33157 mdetpmtr1 33424 mdetpmtr12 33426 mdetlap 33433 zrhf1ker 33576 zrhunitpreima 33579 elzrhunit 33580 qqhval2lem 33582 qqhf 33587 qqhghm 33589 qqhrhm 33590 qqhnm 33591 aks6d1c1p2 41580 aks6d1c1p3 41581 aks6d1c1 41587 hashscontpowcl 41591 hashscontpow 41593 aks6d1c4 41595 aks6d1c2 41601 aks6d1c5lem0 41606 aks6d1c5lem1 41607 aks6d1c5lem3 41608 aks6d1c5lem2 41609 aks6d1c5 41610 aks6d1c6lem1 41642 aks6d1c6lem3 41644 aks6d1c6lem5 41649 aks6d1c7lem1 41652 |
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