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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrhf1ker | Structured version Visualization version GIF version |
Description: The kernel of the homomorphism from the integers to a ring, if it is injective. (Contributed by Thierry Arnoux, 26-Oct-2017.) (Revised by Thierry Arnoux, 23-May-2023.) |
Ref | Expression |
---|---|
zrhker.0 | ⊢ 𝐵 = (Base‘𝑅) |
zrhker.1 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
zrhker.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
zrhf1ker | ⊢ (𝑅 ∈ Ring → (𝐿:ℤ–1-1→𝐵 ↔ (◡𝐿 “ { 0 }) = {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhker.1 | . . 3 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
2 | 1 | zrhrhm 20741 | . 2 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) |
3 | rhmghm 19997 | . 2 ⊢ (𝐿 ∈ (ℤring RingHom 𝑅) → 𝐿 ∈ (ℤring GrpHom 𝑅)) | |
4 | zringbas 20704 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
5 | zrhker.0 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
6 | zring0 20708 | . . 3 ⊢ 0 = (0g‘ℤring) | |
7 | zrhker.2 | . . 3 ⊢ 0 = (0g‘𝑅) | |
8 | 4, 5, 6, 7 | kerf1ghm 20015 | . 2 ⊢ (𝐿 ∈ (ℤring GrpHom 𝑅) → (𝐿:ℤ–1-1→𝐵 ↔ (◡𝐿 “ { 0 }) = {0})) |
9 | 2, 3, 8 | 3syl 18 | 1 ⊢ (𝑅 ∈ Ring → (𝐿:ℤ–1-1→𝐵 ↔ (◡𝐿 “ { 0 }) = {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1537 ∈ wcel 2101 {csn 4564 ◡ccnv 5590 “ cima 5594 –1-1→wf1 6444 ‘cfv 6447 (class class class)co 7295 0cc0 10899 ℤcz 12347 Basecbs 16940 0gc0g 17178 GrpHom cghm 18859 Ringcrg 19811 RingHom crh 19984 ℤringczring 20698 ℤRHomczrh 20729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-addf 10978 ax-mulf 10979 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-seq 13750 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-starv 17005 df-tset 17009 df-ple 17010 df-ds 17012 df-unif 17013 df-0g 17180 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-mhm 18458 df-grp 18608 df-minusg 18609 df-sbg 18610 df-mulg 18729 df-subg 18780 df-ghm 18860 df-cmn 19416 df-mgp 19749 df-ur 19766 df-ring 19813 df-cring 19814 df-rnghom 19987 df-subrg 20050 df-cnfld 20626 df-zring 20699 df-zrh 20733 |
This theorem is referenced by: zrhker 31955 qqhre 31998 |
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