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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 513 | . 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 2733 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
9 | eqid 2733 | . . . . . 6 β’ (.rβπ ) = (.rβπ ) | |
10 | eqid 2733 | . . . . . 6 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
11 | idlmulssprm.1 | . . . . . 6 β’ Γ = (LSSumβ(mulGrpβπ )) | |
12 | eqid 2733 | . . . . . . . . 9 β’ (LIdealβπ ) = (LIdealβπ ) | |
13 | 8, 12 | lidlss 20696 | . . . . . . . 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 20696 | . . . . . . . 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 486 | . . . . . 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 32223 | . . . . 5 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β (π₯(.rβπ )π¦) β (πΌ Γ π½)) |
22 | 7, 21 | sseldd 3946 | . . . 4 β’ (((π β§ π₯ β πΌ) β§ π¦ β π½) β (π₯(.rβπ )π¦) β π) |
23 | 22 | anasss 468 | . . 3 β’ ((π β§ (π₯ β πΌ β§ π¦ β π½)) β (π₯(.rβπ )π¦) β π) |
24 | 23 | ralrimivva 3194 | . 2 β’ (π β βπ₯ β πΌ βπ¦ β π½ (π₯(.rβπ )π¦) β π) |
25 | 8, 9 | prmidl 32260 | . 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 397 β¨ wo 846 = wceq 1542 β wcel 2107 βwral 3061 β wss 3911 βcfv 6497 (class class class)co 7358 Basecbs 17088 .rcmulr 17139 LSSumclsm 19421 mulGrpcmgp 19901 Ringcrg 19969 LIdealclidl 20647 PrmIdealcprmidl 32255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-sca 17154 df-vsca 17155 df-ip 17156 df-lsm 19423 df-mgp 19902 df-lss 20408 df-sra 20649 df-rgmod 20650 df-lidl 20651 df-prmidl 32256 |
This theorem is referenced by: zarclsun 32508 |
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