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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 2724 | . . . 4 β’ (.gβπ ) = (.gβπ ) | |
3 | eqid 2724 | . . . 4 β’ (1rβπ ) = (1rβπ ) | |
4 | 1, 2, 3 | zrhval2 21384 | . . 3 β’ (π β Ring β πΏ = (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ )))) |
5 | f1eq1 6773 | . . 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 20139 | . . 3 β’ (π β Ring β π β Grp) | |
8 | zrhker.0 | . . . 4 β’ π΅ = (Baseβπ ) | |
9 | 8, 3 | ringidcl 20161 | . . 3 β’ (π β Ring β (1rβπ ) β π΅) |
10 | eqid 2724 | . . . 4 β’ (odβπ ) = (odβπ ) | |
11 | eqid 2724 | . . . 4 β’ (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))) = (π₯ β β€ β¦ (π₯(.gβπ )(1rβπ ))) | |
12 | 8, 10, 2, 11 | odf1 19478 | . . 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 2724 | . . . . 5 β’ (chrβπ ) = (chrβπ ) | |
15 | 10, 3, 14 | chrval 21403 | . . . 4 β’ ((odβπ )β(1rβπ )) = (chrβπ ) |
16 | 15 | eqeq1i 2729 | . . 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 1533 β wcel 2098 β¦ cmpt 5222 β1-1βwf1 6531 βcfv 6534 (class class class)co 7402 0cc0 11107 β€cz 12557 Basecbs 17149 0gc0g 17390 Grpcgrp 18859 .gcmg 18991 odcod 19440 1rcur 20082 Ringcrg 20134 β€RHomczrh 21375 chrcchr 21377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-rp 12976 df-fz 13486 df-fl 13758 df-mod 13836 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-dvds 16201 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-ghm 19135 df-od 19444 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-cnfld 21235 df-zring 21323 df-zrh 21379 df-chr 21381 |
This theorem is referenced by: zrhker 33477 qqhre 33520 |
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