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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 1191 | . 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 1192 | . 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 33188 | . . . . 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 475 | . . . 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 1141 | . . 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 479 | . 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 1193 | . 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 7435 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π Β· π) = (πΌ Β· π½)) | |
| 11 | 10 | eleq1d 2814 | . . . . 5 β’ ((π = πΌ β§ π = π½) β ((π Β· π) β π β (πΌ Β· π½) β π)) |
| 12 | simpl 481 | . . . . . . 7 β’ ((π = πΌ β§ π = π½) β π = πΌ) | |
| 13 | 12 | eleq1d 2814 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π β π β πΌ β π)) |
| 14 | simpr 483 | . . . . . . 7 β’ ((π = πΌ β§ π = π½) β π = π½) | |
| 15 | 14 | eleq1d 2814 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π β π β π½ β π)) |
| 16 | 13, 15 | orbi12d 916 | . . . . 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 343 | . . . 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 3621 | . . 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 416 | . 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 394 β¨ wo 845 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 βwral 3058 βcfv 6553 (class class class)co 7426 Basecbs 17187 .rcmulr 17241 CRingccrg 20181 LIdealclidl 21109 PrmIdealcprmidl 33176 |
| 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
| 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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-sra 21065 df-rgmod 21066 df-lidl 21111 df-rsp 21112 df-prmidl 33177 |
| This theorem is referenced by: rhmpreimaprmidl 33192 rsprprmprmidlb 33265 rprmirredb 33271 |
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