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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 | β’ π΅ = (Baseβπ ) |
zrhker.1 | β’ πΏ = (β€RHomβπ ) |
zrhker.2 | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
zrhchr | β’ (π β Ring β ((chrβπ ) = 0 β πΏ:β€β1-1βπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhker.1 | . . . 4 β’ πΏ = (β€RHomβπ ) | |
2 | eqid 2728 | . . . 4 β’ (.gβπ ) = (.gβπ ) | |
3 | eqid 2728 | . . . 4 β’ (1rβπ ) = (1rβπ ) | |
4 | 1, 2, 3 | zrhval2 21434 | . . 3 β’ (π β Ring β πΏ = (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ )))) |
5 | f1eq1 6788 | . . 3 β’ (πΏ = (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))) β (πΏ:β€β1-1βπ΅ β (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))):β€β1-1βπ΅)) | |
6 | 4, 5 | syl 17 | . 2 β’ (π β Ring β (πΏ:β€β1-1βπ΅ β (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))):β€β1-1βπ΅)) |
7 | ringgrp 20178 | . . 3 β’ (π β Ring β π β Grp) | |
8 | zrhker.0 | . . . 4 β’ π΅ = (Baseβπ ) | |
9 | 8, 3 | ringidcl 20202 | . . 3 β’ (π β Ring β (1rβπ ) β π΅) |
10 | eqid 2728 | . . . 4 β’ (odβπ ) = (odβπ ) | |
11 | eqid 2728 | . . . 4 β’ (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))) = (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))) | |
12 | 8, 10, 2, 11 | odf1 19517 | . . 3 β’ ((π β Grp β§ (1rβπ ) β π΅) β (((odβπ )β(1rβπ )) = 0 β (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))):β€β1-1βπ΅)) |
13 | 7, 9, 12 | syl2anc 583 | . 2 β’ (π β Ring β (((odβπ )β(1rβπ )) = 0 β (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))):β€β1-1βπ΅)) |
14 | eqid 2728 | . . . . 5 β’ (chrβπ ) = (chrβπ ) | |
15 | 10, 3, 14 | chrval 21453 | . . . 4 β’ ((odβπ )β(1rβπ )) = (chrβπ ) |
16 | 15 | eqeq1i 2733 | . . 3 β’ (((odβπ )β(1rβπ )) = 0 β (chrβπ ) = 0) |
17 | 16 | a1i 11 | . 2 β’ (π β Ring β (((odβπ )β(1rβπ )) = 0 β (chrβπ ) = 0)) |
18 | 6, 13, 17 | 3bitr2rd 308 | 1 β’ (π β Ring β ((chrβπ ) = 0 β πΏ:β€β1-1βπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1534 β wcel 2099 β¦ cmpt 5231 β1-1βwf1 6545 βcfv 6548 (class class class)co 7420 0cc0 11139 β€cz 12589 Basecbs 17180 0gc0g 17421 Grpcgrp 18890 .gcmg 19023 odcod 19479 1rcur 20121 Ringcrg 20173 β€RHomczrh 21425 chrcchr 21427 |
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-pre-sup 11217 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-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 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-rp 13008 df-fz 13518 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-dvds 16232 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-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-od 19483 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 df-chr 21431 |
This theorem is referenced by: zrhker 33578 qqhre 33621 |
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