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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrhchr | Structured version Visualization version GIF version |
Description: The kernel of the homomorphism from the integers to a ring is injective if and only if the ring has characteristic 0 . (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
zrhker.0 | ⊢ 𝐵 = (Base‘𝑅) |
zrhker.1 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
zrhker.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
zrhchr | ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ 𝐿:ℤ–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhker.1 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
2 | eqid 2734 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
3 | eqid 2734 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 1, 2, 3 | zrhval2 21537 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐿 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅)))) |
5 | f1eq1 6811 | . . 3 ⊢ (𝐿 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))) → (𝐿:ℤ–1-1→𝐵 ↔ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))):ℤ–1-1→𝐵)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (𝐿:ℤ–1-1→𝐵 ↔ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))):ℤ–1-1→𝐵)) |
7 | ringgrp 20260 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
8 | zrhker.0 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
9 | 8, 3 | ringidcl 20284 | . . 3 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
10 | eqid 2734 | . . . 4 ⊢ (od‘𝑅) = (od‘𝑅) | |
11 | eqid 2734 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))) = (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))) | |
12 | 8, 10, 2, 11 | odf1 19599 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ 𝐵) → (((od‘𝑅)‘(1r‘𝑅)) = 0 ↔ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))):ℤ–1-1→𝐵)) |
13 | 7, 9, 12 | syl2anc 583 | . 2 ⊢ (𝑅 ∈ Ring → (((od‘𝑅)‘(1r‘𝑅)) = 0 ↔ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))):ℤ–1-1→𝐵)) |
14 | eqid 2734 | . . . . 5 ⊢ (chr‘𝑅) = (chr‘𝑅) | |
15 | 10, 3, 14 | chrval 21556 | . . . 4 ⊢ ((od‘𝑅)‘(1r‘𝑅)) = (chr‘𝑅) |
16 | 15 | eqeq1i 2739 | . . 3 ⊢ (((od‘𝑅)‘(1r‘𝑅)) = 0 ↔ (chr‘𝑅) = 0) |
17 | 16 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → (((od‘𝑅)‘(1r‘𝑅)) = 0 ↔ (chr‘𝑅) = 0)) |
18 | 6, 13, 17 | 3bitr2rd 308 | 1 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ 𝐿:ℤ–1-1→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2103 ↦ cmpt 5252 –1-1→wf1 6569 ‘cfv 6572 (class class class)co 7445 0cc0 11180 ℤcz 12635 Basecbs 17253 0gc0g 17494 Grpcgrp 18968 .gcmg 19102 odcod 19561 1rcur 20203 Ringcrg 20255 ℤRHomczrh 21528 chrcchr 21530 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259 ax-mulf 11260 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-sup 9507 df-inf 9508 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-rp 13054 df-fz 13564 df-fl 13839 df-mod 13917 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-dvds 16297 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-0g 17496 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mhm 18813 df-grp 18971 df-minusg 18972 df-sbg 18973 df-mulg 19103 df-subg 19158 df-ghm 19248 df-od 19565 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-rhm 20493 df-subrng 20567 df-subrg 20592 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-chr 21534 |
This theorem is referenced by: zrhker 33915 qqhre 33958 |
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