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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlmulssprm | Structured version Visualization version GIF version |
Description: Let π be a prime ideal containing the product (πΌ Γ π½) of two ideals πΌ and π½. Then πΌ β π or π½ β π. (Contributed by Thierry Arnoux, 13-Apr-2024.) |
Ref | Expression |
---|---|
idlmulssprm.1 | β’ Γ = (LSSumβ(mulGrpβπ )) |
idlmulssprm.2 | β’ (π β π β Ring) |
idlmulssprm.3 | β’ (π β π β (PrmIdealβπ )) |
idlmulssprm.4 | β’ (π β πΌ β (LIdealβπ )) |
idlmulssprm.5 | β’ (π β π½ β (LIdealβπ )) |
idlmulssprm.6 | β’ (π β (πΌ Γ π½) β π) |
Ref | Expression |
---|---|
idlmulssprm | β’ (π β (πΌ β π β¨ π½ β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlmulssprm.2 | . 2 β’ (π β π β Ring) | |
2 | idlmulssprm.3 | . 2 β’ (π β π β (PrmIdealβπ )) | |
3 | idlmulssprm.4 | . . 3 β’ (π β πΌ β (LIdealβπ )) | |
4 | idlmulssprm.5 | . . 3 β’ (π β π½ β (LIdealβπ )) | |
5 | 3, 4 | jca 510 | . 2 β’ (π β (πΌ β (LIdealβπ ) β§ π½ β (LIdealβπ ))) |
6 | idlmulssprm.6 | . . . . . 6 β’ (π β (πΌ Γ π½) β π) | |
7 | 6 | ad2antrr 724 | . . . . 5 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β (πΌ Γ π½) β π) |
8 | eqid 2725 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
9 | eqid 2725 | . . . . . 6 β’ (.rβπ ) = (.rβπ ) | |
10 | eqid 2725 | . . . . . 6 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
11 | idlmulssprm.1 | . . . . . 6 β’ Γ = (LSSumβ(mulGrpβπ )) | |
12 | eqid 2725 | . . . . . . . . 9 β’ (LIdealβπ ) = (LIdealβπ ) | |
13 | 8, 12 | lidlss 21107 | . . . . . . . 8 β’ (πΌ β (LIdealβπ ) β πΌ β (Baseβπ )) |
14 | 3, 13 | syl 17 | . . . . . . 7 β’ (π β πΌ β (Baseβπ )) |
15 | 14 | ad2antrr 724 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β πΌ β (Baseβπ )) |
16 | 8, 12 | lidlss 21107 | . . . . . . . 8 β’ (π½ β (LIdealβπ ) β π½ β (Baseβπ )) |
17 | 4, 16 | syl 17 | . . . . . . 7 β’ (π β π½ β (Baseβπ )) |
18 | 17 | ad2antrr 724 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β π½ β (Baseβπ )) |
19 | simplr 767 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β π₯ β πΌ) | |
20 | simpr 483 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β π¦ β π½) | |
21 | 8, 9, 10, 11, 15, 18, 19, 20 | elringlsmd 33148 | . . . . 5 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β (π₯(.rβπ )π¦) β (πΌ Γ π½)) |
22 | 7, 21 | sseldd 3974 | . . . 4 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β (π₯(.rβπ )π¦) β π) |
23 | 22 | anasss 465 | . . 3 β’ ((π β§ (π₯ β πΌ β§ π¦ β π½)) β (π₯(.rβπ )π¦) β π) |
24 | 23 | ralrimivva 3191 | . 2 β’ (π β βπ₯ β πΌ βπ¦ β π½ (π₯(.rβπ )π¦) β π) |
25 | 8, 9 | prmidl 33201 | . 2 β’ ((((π β Ring β§ π β (PrmIdealβπ )) β§ (πΌ β (LIdealβπ ) β§ π½ β (LIdealβπ ))) β§ βπ₯ β πΌ βπ¦ β π½ (π₯(.rβπ )π¦) β π) β (πΌ β π β¨ π½ β π)) |
26 | 1, 2, 5, 24, 25 | syl1111anc 838 | 1 β’ (π β (πΌ β π β¨ π½ β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β¨ wo 845 = wceq 1533 β wcel 2098 βwral 3051 β wss 3941 βcfv 6543 (class class class)co 7413 Basecbs 17174 .rcmulr 17228 LSSumclsm 19588 mulGrpcmgp 20073 Ringcrg 20172 LIdealclidl 21101 PrmIdealcprmidl 33196 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-sca 17243 df-vsca 17244 df-ip 17245 df-lsm 19590 df-mgp 20074 df-lss 20815 df-sra 21057 df-rgmod 21058 df-lidl 21103 df-prmidl 33197 |
This theorem is referenced by: zarclsun 33524 |
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