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 | ⊢ 𝐶 = (chr‘𝑅) |
dvdschrmulg.2 | ⊢ 𝐵 = (Base‘𝑅) |
dvdschrmulg.3 | ⊢ · = (.g‘𝑅) |
dvdschrmulg.4 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
dvdschrmulg | ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → (𝑁 · 𝐴) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → 𝑅 ∈ Ring) | |
2 | dvdszrcl 15612 | . . . . 5 ⊢ (𝐶 ∥ 𝑁 → (𝐶 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
3 | 2 | simprd 498 | . . . 4 ⊢ (𝐶 ∥ 𝑁 → 𝑁 ∈ ℤ) |
4 | 3 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → 𝑁 ∈ ℤ) |
5 | dvdschrmulg.2 | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
6 | eqid 2821 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 5, 6 | ringidcl 19318 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
8 | 1, 7 | syl 17 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → (1r‘𝑅) ∈ 𝐵) |
9 | simp3 1134 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
10 | dvdschrmulg.3 | . . . 4 ⊢ · = (.g‘𝑅) | |
11 | eqid 2821 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | 5, 10, 11 | mulgass2 19351 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ (1r‘𝑅) ∈ 𝐵 ∧ 𝐴 ∈ 𝐵)) → ((𝑁 · (1r‘𝑅))(.r‘𝑅)𝐴) = (𝑁 · ((1r‘𝑅)(.r‘𝑅)𝐴))) |
13 | 1, 4, 8, 9, 12 | syl13anc 1368 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → ((𝑁 · (1r‘𝑅))(.r‘𝑅)𝐴) = (𝑁 · ((1r‘𝑅)(.r‘𝑅)𝐴))) |
14 | ringgrp 19302 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
15 | 1, 14 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → 𝑅 ∈ Grp) |
16 | eqid 2821 | . . . . . . 7 ⊢ (od‘𝑅) = (od‘𝑅) | |
17 | dvdschrmulg.1 | . . . . . . 7 ⊢ 𝐶 = (chr‘𝑅) | |
18 | 16, 6, 17 | chrval 20672 | . . . . . 6 ⊢ ((od‘𝑅)‘(1r‘𝑅)) = 𝐶 |
19 | simp2 1133 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → 𝐶 ∥ 𝑁) | |
20 | 18, 19 | eqbrtrid 5101 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → ((od‘𝑅)‘(1r‘𝑅)) ∥ 𝑁) |
21 | dvdschrmulg.4 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
22 | 5, 16, 10, 21 | oddvdsi 18676 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ 𝐵 ∧ ((od‘𝑅)‘(1r‘𝑅)) ∥ 𝑁) → (𝑁 · (1r‘𝑅)) = 0 ) |
23 | 15, 8, 20, 22 | syl3anc 1367 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → (𝑁 · (1r‘𝑅)) = 0 ) |
24 | 23 | oveq1d 7171 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → ((𝑁 · (1r‘𝑅))(.r‘𝑅)𝐴) = ( 0 (.r‘𝑅)𝐴)) |
25 | 5, 11, 21 | ringlz 19337 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵) → ( 0 (.r‘𝑅)𝐴) = 0 ) |
26 | 25 | 3adant2 1127 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → ( 0 (.r‘𝑅)𝐴) = 0 ) |
27 | 24, 26 | eqtrd 2856 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → ((𝑁 · (1r‘𝑅))(.r‘𝑅)𝐴) = 0 ) |
28 | 5, 11, 6 | ringlidm 19321 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵) → ((1r‘𝑅)(.r‘𝑅)𝐴) = 𝐴) |
29 | 28 | 3adant2 1127 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → ((1r‘𝑅)(.r‘𝑅)𝐴) = 𝐴) |
30 | 29 | oveq2d 7172 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → (𝑁 · ((1r‘𝑅)(.r‘𝑅)𝐴)) = (𝑁 · 𝐴)) |
31 | 13, 27, 30 | 3eqtr3rd 2865 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → (𝑁 · 𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℤcz 11982 ∥ cdvds 15607 Basecbs 16483 .rcmulr 16566 0gc0g 16713 Grpcgrp 18103 .gcmg 18224 odcod 18652 1rcur 19251 Ringcrg 19297 chrcchr 20649 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-od 18656 df-mgp 19240 df-ur 19252 df-ring 19299 df-chr 20653 |
This theorem is referenced by: freshmansdream 30859 |
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