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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 511 | . 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 725 | . . . . 5 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β (πΌ Γ π½) β π) |
8 | eqid 2727 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
9 | eqid 2727 | . . . . . 6 β’ (.rβπ ) = (.rβπ ) | |
10 | eqid 2727 | . . . . . 6 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
11 | idlmulssprm.1 | . . . . . 6 β’ Γ = (LSSumβ(mulGrpβπ )) | |
12 | eqid 2727 | . . . . . . . . 9 β’ (LIdealβπ ) = (LIdealβπ ) | |
13 | 8, 12 | lidlss 21090 | . . . . . . . 8 β’ (πΌ β (LIdealβπ ) β πΌ β (Baseβπ )) |
14 | 3, 13 | syl 17 | . . . . . . 7 β’ (π β πΌ β (Baseβπ )) |
15 | 14 | ad2antrr 725 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β πΌ β (Baseβπ )) |
16 | 8, 12 | lidlss 21090 | . . . . . . . 8 β’ (π½ β (LIdealβπ ) β π½ β (Baseβπ )) |
17 | 4, 16 | syl 17 | . . . . . . 7 β’ (π β π½ β (Baseβπ )) |
18 | 17 | ad2antrr 725 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β π½ β (Baseβπ )) |
19 | simplr 768 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β π₯ β πΌ) | |
20 | simpr 484 | . . . . . 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 33030 | . . . . 5 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β (π₯(.rβπ )π¦) β (πΌ Γ π½)) |
22 | 7, 21 | sseldd 3979 | . . . 4 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β (π₯(.rβπ )π¦) β π) |
23 | 22 | anasss 466 | . . 3 β’ ((π β§ (π₯ β πΌ β§ π¦ β π½)) β (π₯(.rβπ )π¦) β π) |
24 | 23 | ralrimivva 3195 | . 2 β’ (π β βπ₯ β πΌ βπ¦ β π½ (π₯(.rβπ )π¦) β π) |
25 | 8, 9 | prmidl 33078 | . 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 839 | 1 β’ (π β (πΌ β π β¨ π½ β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β¨ wo 846 = wceq 1534 β wcel 2099 βwral 3056 β wss 3944 βcfv 6542 (class class class)co 7414 Basecbs 17165 .rcmulr 17219 LSSumclsm 19573 mulGrpcmgp 20058 Ringcrg 20157 LIdealclidl 21084 PrmIdealcprmidl 33073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-plusg 17231 df-sca 17234 df-vsca 17235 df-ip 17236 df-lsm 19575 df-mgp 20059 df-lss 20798 df-sra 21040 df-rgmod 21041 df-lidl 21086 df-prmidl 33074 |
This theorem is referenced by: zarclsun 33394 |
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