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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdsrspss | Structured version Visualization version GIF version |
Description: In a ring, an element π divides π iff the ideal generated by π is a subset of the ideal generated by π (Contributed by Thierry Arnoux, 22-Mar-2025.) |
Ref | Expression |
---|---|
dvdsrspss.b | β’ π΅ = (Baseβπ ) |
dvdsrspss.k | β’ πΎ = (RSpanβπ ) |
dvdsrspss.d | β’ β₯ = (β₯rβπ ) |
dvdsrspss.x | β’ (π β π β π΅) |
dvdsrspss.y | β’ (π β π β π΅) |
dvdsrspss.r | β’ (π β π β Ring) |
Ref | Expression |
---|---|
dvdsrspss | β’ (π β (π β₯ π β (πΎβ{π}) β (πΎβ{π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsrspss.b | . . . 4 β’ π΅ = (Baseβπ ) | |
2 | dvdsrspss.d | . . . 4 β’ β₯ = (β₯rβπ ) | |
3 | eqid 2725 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
4 | 1, 2, 3 | dvdsr 20305 | . . 3 β’ (π β₯ π β (π β π΅ β§ βπ‘ β π΅ (π‘(.rβπ )π) = π)) |
5 | dvdsrspss.x | . . . 4 β’ (π β π β π΅) | |
6 | 5 | biantrurd 531 | . . 3 β’ (π β (βπ‘ β π΅ (π‘(.rβπ )π) = π β (π β π΅ β§ βπ‘ β π΅ (π‘(.rβπ )π) = π))) |
7 | 4, 6 | bitr4id 289 | . 2 β’ (π β (π β₯ π β βπ‘ β π΅ (π‘(.rβπ )π) = π)) |
8 | dvdsrspss.r | . . . 4 β’ (π β π β Ring) | |
9 | dvdsrspss.k | . . . . 5 β’ πΎ = (RSpanβπ ) | |
10 | 1, 3, 9 | rspsnel 33131 | . . . 4 β’ ((π β Ring β§ π β π΅) β (π β (πΎβ{π}) β βπ‘ β π΅ π = (π‘(.rβπ )π))) |
11 | 8, 5, 10 | syl2anc 582 | . . 3 β’ (π β (π β (πΎβ{π}) β βπ‘ β π΅ π = (π‘(.rβπ )π))) |
12 | eqcom 2732 | . . . 4 β’ ((π‘(.rβπ )π) = π β π = (π‘(.rβπ )π)) | |
13 | 12 | rexbii 3084 | . . 3 β’ (βπ‘ β π΅ (π‘(.rβπ )π) = π β βπ‘ β π΅ π = (π‘(.rβπ )π)) |
14 | 11, 13 | bitr4di 288 | . 2 β’ (π β (π β (πΎβ{π}) β βπ‘ β π΅ (π‘(.rβπ )π) = π)) |
15 | 8 | adantr 479 | . . . 4 β’ ((π β§ π β (πΎβ{π})) β π β Ring) |
16 | 5 | snssd 4808 | . . . . . 6 β’ (π β {π} β π΅) |
17 | eqid 2725 | . . . . . . 7 β’ (LIdealβπ ) = (LIdealβπ ) | |
18 | 9, 1, 17 | rspcl 21135 | . . . . . 6 β’ ((π β Ring β§ {π} β π΅) β (πΎβ{π}) β (LIdealβπ )) |
19 | 8, 16, 18 | syl2anc 582 | . . . . 5 β’ (π β (πΎβ{π}) β (LIdealβπ )) |
20 | 19 | adantr 479 | . . . 4 β’ ((π β§ π β (πΎβ{π})) β (πΎβ{π}) β (LIdealβπ )) |
21 | simpr 483 | . . . . 5 β’ ((π β§ π β (πΎβ{π})) β π β (πΎβ{π})) | |
22 | 21 | snssd 4808 | . . . 4 β’ ((π β§ π β (πΎβ{π})) β {π} β (πΎβ{π})) |
23 | 9, 17 | rspssp 21139 | . . . 4 β’ ((π β Ring β§ (πΎβ{π}) β (LIdealβπ ) β§ {π} β (πΎβ{π})) β (πΎβ{π}) β (πΎβ{π})) |
24 | 15, 20, 22, 23 | syl3anc 1368 | . . 3 β’ ((π β§ π β (πΎβ{π})) β (πΎβ{π}) β (πΎβ{π})) |
25 | simpr 483 | . . . 4 β’ ((π β§ (πΎβ{π}) β (πΎβ{π})) β (πΎβ{π}) β (πΎβ{π})) | |
26 | dvdsrspss.y | . . . . . 6 β’ (π β π β π΅) | |
27 | 26 | snssd 4808 | . . . . . . 7 β’ (π β {π} β π΅) |
28 | 9, 1 | rspssid 21136 | . . . . . . 7 β’ ((π β Ring β§ {π} β π΅) β {π} β (πΎβ{π})) |
29 | 8, 27, 28 | syl2anc 582 | . . . . . 6 β’ (π β {π} β (πΎβ{π})) |
30 | snssg 4783 | . . . . . . 7 β’ (π β π΅ β (π β (πΎβ{π}) β {π} β (πΎβ{π}))) | |
31 | 30 | biimpar 476 | . . . . . 6 β’ ((π β π΅ β§ {π} β (πΎβ{π})) β π β (πΎβ{π})) |
32 | 26, 29, 31 | syl2anc 582 | . . . . 5 β’ (π β π β (πΎβ{π})) |
33 | 32 | adantr 479 | . . . 4 β’ ((π β§ (πΎβ{π}) β (πΎβ{π})) β π β (πΎβ{π})) |
34 | 25, 33 | sseldd 3973 | . . 3 β’ ((π β§ (πΎβ{π}) β (πΎβ{π})) β π β (πΎβ{π})) |
35 | 24, 34 | impbida 799 | . 2 β’ (π β (π β (πΎβ{π}) β (πΎβ{π}) β (πΎβ{π}))) |
36 | 7, 14, 35 | 3bitr2d 306 | 1 β’ (π β (π β₯ π β (πΎβ{π}) β (πΎβ{π}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βwrex 3060 β wss 3939 {csn 4624 class class class wbr 5143 βcfv 6543 (class class class)co 7416 Basecbs 17179 .rcmulr 17233 Ringcrg 20177 β₯rcdsr 20297 LIdealclidl 21106 RSpancrsp 21107 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-mgp 20079 df-ur 20126 df-ring 20179 df-dvdsr 20300 df-subrg 20512 df-lmod 20749 df-lss 20820 df-lsp 20860 df-sra 21062 df-rgmod 21063 df-lidl 21108 df-rsp 21109 |
This theorem is referenced by: rspsnasso 33140 |
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