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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdschrmulg | Structured version Visualization version GIF version | ||
| Description: In a ring, any multiple of the characteristics annihilates all elements. (Contributed by Thierry Arnoux, 6-Sep-2016.) |
| Ref | Expression |
|---|---|
| dvdschrmulg.1 | β’ πΆ = (chrβπ ) |
| dvdschrmulg.2 | β’ π΅ = (Baseβπ ) |
| dvdschrmulg.3 | β’ Β· = (.gβπ ) |
| dvdschrmulg.4 | β’ 0 = (0gβπ ) |
| Ref | Expression |
|---|---|
| dvdschrmulg | β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β (π Β· π΄) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1135 | . . 3 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β π β Ring) | |
| 2 | dvdszrcl 16207 | . . . . 5 β’ (πΆ β₯ π β (πΆ β β€ β§ π β β€)) | |
| 3 | 2 | simprd 495 | . . . 4 β’ (πΆ β₯ π β π β β€) |
| 4 | 3 | 3ad2ant2 1133 | . . 3 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β π β β€) |
| 5 | dvdschrmulg.2 | . . . . 5 β’ π΅ = (Baseβπ ) | |
| 6 | eqid 2731 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
| 7 | 5, 6 | ringidcl 20155 | . . . 4 β’ (π β Ring β (1rβπ ) β π΅) |
| 8 | 1, 7 | syl 17 | . . 3 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β (1rβπ ) β π΅) |
| 9 | simp3 1137 | . . 3 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β π΄ β π΅) | |
| 10 | dvdschrmulg.3 | . . . 4 β’ Β· = (.gβπ ) | |
| 11 | eqid 2731 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
| 12 | 5, 10, 11 | mulgass2 20198 | . . 3 β’ ((π β Ring β§ (π β β€ β§ (1rβπ ) β π΅ β§ π΄ β π΅)) β ((π Β· (1rβπ ))(.rβπ )π΄) = (π Β· ((1rβπ )(.rβπ )π΄))) |
| 13 | 1, 4, 8, 9, 12 | syl13anc 1371 | . 2 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β ((π Β· (1rβπ ))(.rβπ )π΄) = (π Β· ((1rβπ )(.rβπ )π΄))) |
| 14 | ringgrp 20133 | . . . . . 6 β’ (π β Ring β π β Grp) | |
| 15 | 1, 14 | syl 17 | . . . . 5 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β π β Grp) |
| 16 | eqid 2731 | . . . . . . 7 β’ (odβπ ) = (odβπ ) | |
| 17 | dvdschrmulg.1 | . . . . . . 7 β’ πΆ = (chrβπ ) | |
| 18 | 16, 6, 17 | chrval 21297 | . . . . . 6 β’ ((odβπ )β(1rβπ )) = πΆ |
| 19 | simp2 1136 | . . . . . 6 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β πΆ β₯ π) | |
| 20 | 18, 19 | eqbrtrid 5183 | . . . . 5 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β ((odβπ )β(1rβπ )) β₯ π) |
| 21 | dvdschrmulg.4 | . . . . . 6 β’ 0 = (0gβπ ) | |
| 22 | 5, 16, 10, 21 | oddvdsi 19458 | . . . . 5 β’ ((π β Grp β§ (1rβπ ) β π΅ β§ ((odβπ )β(1rβπ )) β₯ π) β (π Β· (1rβπ )) = 0 ) |
| 23 | 15, 8, 20, 22 | syl3anc 1370 | . . . 4 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β (π Β· (1rβπ )) = 0 ) |
| 24 | 23 | oveq1d 7427 | . . 3 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β ((π Β· (1rβπ ))(.rβπ )π΄) = ( 0 (.rβπ )π΄)) |
| 25 | 5, 11, 21 | ringlz 20182 | . . . 4 β’ ((π β Ring β§ π΄ β π΅) β ( 0 (.rβπ )π΄) = 0 ) |
| 26 | 25 | 3adant2 1130 | . . 3 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β ( 0 (.rβπ )π΄) = 0 ) |
| 27 | 24, 26 | eqtrd 2771 | . 2 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β ((π Β· (1rβπ ))(.rβπ )π΄) = 0 ) |
| 28 | 5, 11, 6 | ringlidm 20158 | . . . 4 β’ ((π β Ring β§ π΄ β π΅) β ((1rβπ )(.rβπ )π΄) = π΄) |
| 29 | 28 | 3adant2 1130 | . . 3 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β ((1rβπ )(.rβπ )π΄) = π΄) |
| 30 | 29 | oveq2d 7428 | . 2 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β (π Β· ((1rβπ )(.rβπ )π΄)) = (π Β· π΄)) |
| 31 | 13, 27, 30 | 3eqtr3rd 2780 | 1 β’ ((π β Ring β§ πΆ β₯ π β§ π΄ β π΅) β (π Β· π΄) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 class class class wbr 5148 βcfv 6543 (class class class)co 7412 β€cz 12563 β₯ cdvds 16202 Basecbs 17149 .rcmulr 17203 0gc0g 17390 Grpcgrp 18856 .gcmg 18987 odcod 19434 1rcur 20076 Ringcrg 20128 chrcchr 21271 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-sup 9441 df-inf 9442 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-dvds 16203 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-od 19438 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-chr 21275 |
| This theorem is referenced by: freshmansdream 32652 |
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