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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrhunitpreima | Structured version Visualization version GIF version |
Description: The preimage by ℤRHom of the units of a division ring is (ℤ ∖ {0}). (Contributed by Thierry Arnoux, 22-Oct-2017.) |
Ref | Expression |
---|---|
zrhker.0 | ⊢ 𝐵 = (Base‘𝑅) |
zrhker.1 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
zrhker.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
zrhunitpreima | ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhker.0 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2726 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | eqid 2726 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 1, 2, 3 | isdrng 20707 | . . . . 5 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ {(0g‘𝑅)}))) |
5 | 4 | simprbi 495 | . . . 4 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ {(0g‘𝑅)})) |
6 | 5 | imaeq2d 6061 | . . 3 ⊢ (𝑅 ∈ DivRing → (◡𝐿 “ (Unit‘𝑅)) = (◡𝐿 “ (𝐵 ∖ {(0g‘𝑅)}))) |
7 | 6 | adantr 479 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ (Unit‘𝑅)) = (◡𝐿 “ (𝐵 ∖ {(0g‘𝑅)}))) |
8 | drngring 20710 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
9 | zrhker.1 | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
10 | 9 | zrhrhm 21497 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) |
11 | zringbas 21439 | . . . . . 6 ⊢ ℤ = (Base‘ℤring) | |
12 | 11, 1 | rhmf 20463 | . . . . 5 ⊢ (𝐿 ∈ (ℤring RingHom 𝑅) → 𝐿:ℤ⟶𝐵) |
13 | ffun 6723 | . . . . 5 ⊢ (𝐿:ℤ⟶𝐵 → Fun 𝐿) | |
14 | 10, 12, 13 | 3syl 18 | . . . 4 ⊢ (𝑅 ∈ Ring → Fun 𝐿) |
15 | difpreima 7070 | . . . 4 ⊢ (Fun 𝐿 → (◡𝐿 “ (𝐵 ∖ {(0g‘𝑅)})) = ((◡𝐿 “ 𝐵) ∖ (◡𝐿 “ {(0g‘𝑅)}))) | |
16 | 8, 14, 15 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ DivRing → (◡𝐿 “ (𝐵 ∖ {(0g‘𝑅)})) = ((◡𝐿 “ 𝐵) ∖ (◡𝐿 “ {(0g‘𝑅)}))) |
17 | 16 | adantr 479 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ (𝐵 ∖ {(0g‘𝑅)})) = ((◡𝐿 “ 𝐵) ∖ (◡𝐿 “ {(0g‘𝑅)}))) |
18 | fimacnv 6742 | . . . . 5 ⊢ (𝐿:ℤ⟶𝐵 → (◡𝐿 “ 𝐵) = ℤ) | |
19 | 8, 10, 12, 18 | 4syl 19 | . . . 4 ⊢ (𝑅 ∈ DivRing → (◡𝐿 “ 𝐵) = ℤ) |
20 | 19 | adantr 479 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ 𝐵) = ℤ) |
21 | 1, 9, 3 | zrhker 33805 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ (◡𝐿 “ {(0g‘𝑅)}) = {0})) |
22 | 21 | biimpa 475 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (chr‘𝑅) = 0) → (◡𝐿 “ {(0g‘𝑅)}) = {0}) |
23 | 8, 22 | sylan 578 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ {(0g‘𝑅)}) = {0}) |
24 | 20, 23 | difeq12d 4119 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((◡𝐿 “ 𝐵) ∖ (◡𝐿 “ {(0g‘𝑅)})) = (ℤ ∖ {0})) |
25 | 7, 17, 24 | 3eqtrd 2770 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∖ cdif 3943 {csn 4623 ◡ccnv 5673 “ cima 5677 Fun wfun 6540 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 0cc0 11149 ℤcz 12604 Basecbs 17208 0gc0g 17449 Ringcrg 20212 Unitcui 20333 RingHom crh 20447 DivRingcdr 20703 ℤringczring 21432 ℤRHomczrh 21485 chrcchr 21487 |
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 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 ax-mulf 11229 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-rp 13023 df-fz 13533 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19058 df-subg 19113 df-ghm 19203 df-od 19522 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-rhm 20450 df-subrng 20524 df-subrg 20549 df-drng 20705 df-cnfld 21340 df-zring 21433 df-zrh 21489 df-chr 21491 |
This theorem is referenced by: elzrhunit 33807 qqhval2 33810 |
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