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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > prmidlc | Structured version Visualization version GIF version | ||
| Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| Ref | Expression |
|---|---|
| isprmidlc.1 | β’ π΅ = (Baseβπ ) |
| isprmidlc.2 | β’ Β· = (.rβπ ) |
| Ref | Expression |
|---|---|
| prmidlc | β’ (((π β CRing β§ π β (PrmIdealβπ )) β§ (πΌ β π΅ β§ π½ β π΅ β§ (πΌ Β· π½) β π)) β (πΌ β π β¨ π½ β π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpr1 1194 | . 2 β’ (((π β CRing β§ π β (PrmIdealβπ )) β§ (πΌ β π΅ β§ π½ β π΅ β§ (πΌ Β· π½) β π)) β πΌ β π΅) | |
| 2 | simpr2 1195 | . 2 β’ (((π β CRing β§ π β (PrmIdealβπ )) β§ (πΌ β π΅ β§ π½ β π΅ β§ (πΌ Β· π½) β π)) β π½ β π΅) | |
| 3 | isprmidlc.1 | . . . . . 6 β’ π΅ = (Baseβπ ) | |
| 4 | isprmidlc.2 | . . . . . 6 β’ Β· = (.rβπ ) | |
| 5 | 3, 4 | isprmidlc 32554 | . . . . 5 β’ (π β CRing β (π β (PrmIdealβπ ) β (π β (LIdealβπ ) β§ π β π΅ β§ βπ β π΅ βπ β π΅ ((π Β· π) β π β (π β π β¨ π β π))))) |
| 6 | 5 | biimpa 477 | . . . 4 β’ ((π β CRing β§ π β (PrmIdealβπ )) β (π β (LIdealβπ ) β§ π β π΅ β§ βπ β π΅ βπ β π΅ ((π Β· π) β π β (π β π β¨ π β π)))) |
| 7 | 6 | simp3d 1144 | . . 3 β’ ((π β CRing β§ π β (PrmIdealβπ )) β βπ β π΅ βπ β π΅ ((π Β· π) β π β (π β π β¨ π β π))) |
| 8 | 7 | adantr 481 | . 2 β’ (((π β CRing β§ π β (PrmIdealβπ )) β§ (πΌ β π΅ β§ π½ β π΅ β§ (πΌ Β· π½) β π)) β βπ β π΅ βπ β π΅ ((π Β· π) β π β (π β π β¨ π β π))) |
| 9 | simpr3 1196 | . 2 β’ (((π β CRing β§ π β (PrmIdealβπ )) β§ (πΌ β π΅ β§ π½ β π΅ β§ (πΌ Β· π½) β π)) β (πΌ Β· π½) β π) | |
| 10 | oveq12 7414 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π Β· π) = (πΌ Β· π½)) | |
| 11 | 10 | eleq1d 2818 | . . . . 5 β’ ((π = πΌ β§ π = π½) β ((π Β· π) β π β (πΌ Β· π½) β π)) |
| 12 | simpl 483 | . . . . . . 7 β’ ((π = πΌ β§ π = π½) β π = πΌ) | |
| 13 | 12 | eleq1d 2818 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π β π β πΌ β π)) |
| 14 | simpr 485 | . . . . . . 7 β’ ((π = πΌ β§ π = π½) β π = π½) | |
| 15 | 14 | eleq1d 2818 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π β π β π½ β π)) |
| 16 | 13, 15 | orbi12d 917 | . . . . 5 β’ ((π = πΌ β§ π = π½) β ((π β π β¨ π β π) β (πΌ β π β¨ π½ β π))) |
| 17 | 11, 16 | imbi12d 344 | . . . 4 β’ ((π = πΌ β§ π = π½) β (((π Β· π) β π β (π β π β¨ π β π)) β ((πΌ Β· π½) β π β (πΌ β π β¨ π½ β π)))) |
| 18 | 17 | rspc2gv 3620 | . . 3 β’ ((πΌ β π΅ β§ π½ β π΅) β (βπ β π΅ βπ β π΅ ((π Β· π) β π β (π β π β¨ π β π)) β ((πΌ Β· π½) β π β (πΌ β π β¨ π½ β π)))) |
| 19 | 18 | imp31 418 | . 2 β’ ((((πΌ β π΅ β§ π½ β π΅) β§ βπ β π΅ βπ β π΅ ((π Β· π) β π β (π β π β¨ π β π))) β§ (πΌ Β· π½) β π) β (πΌ β π β¨ π½ β π)) |
| 20 | 1, 2, 8, 9, 19 | syl1111anc 838 | 1 β’ (((π β CRing β§ π β (PrmIdealβπ )) β§ (πΌ β π΅ β§ π½ β π΅ β§ (πΌ Β· π½) β π)) β (πΌ β π β¨ π½ β π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β¨ wo 845 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 βcfv 6540 (class class class)co 7405 Basecbs 17140 .rcmulr 17194 CRingccrg 20050 LIdealclidl 20775 PrmIdealcprmidl 32541 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cmn 19644 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-subrg 20353 df-lmod 20465 df-lss 20535 df-lsp 20575 df-sra 20777 df-rgmod 20778 df-lidl 20779 df-rsp 20780 df-prmidl 32542 |
| This theorem is referenced by: rhmpreimaprmidl 32558 |
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