![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 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 πΌ β π΅) | |
2 | simpr2 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 π½ β π΅) | |
3 | isprmidlc.1 | . . . . . 6 β’ π΅ = (Baseβπ ) | |
4 | isprmidlc.2 | . . . . . 6 β’ Β· = (.rβπ ) | |
5 | 3, 4 | isprmidlc 32268 | . . . . 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 478 | . . . 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 1145 | . . 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 482 | . 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 1197 | . 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 7367 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π Β· π) = (πΌ Β· π½)) | |
11 | 10 | eleq1d 2819 | . . . . 5 β’ ((π = πΌ β§ π = π½) β ((π Β· π) β π β (πΌ Β· π½) β π)) |
12 | simpl 484 | . . . . . . 7 β’ ((π = πΌ β§ π = π½) β π = πΌ) | |
13 | 12 | eleq1d 2819 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π β π β πΌ β π)) |
14 | simpr 486 | . . . . . . 7 β’ ((π = πΌ β§ π = π½) β π = π½) | |
15 | 14 | eleq1d 2819 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π β π β π½ β π)) |
16 | 13, 15 | orbi12d 918 | . . . . 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 345 | . . . 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 3588 | . . 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 419 | . 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 839 | 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 397 β¨ wo 846 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 βcfv 6497 (class class class)co 7358 Basecbs 17088 .rcmulr 17139 CRingccrg 19970 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-rmo 3352 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-int 4909 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-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-cmn 19569 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-subrg 20234 df-lmod 20338 df-lss 20408 df-lsp 20448 df-sra 20649 df-rgmod 20650 df-lidl 20651 df-rsp 20652 df-prmidl 32256 |
This theorem is referenced by: rhmpreimaprmidl 32272 |
Copyright terms: Public domain | W3C validator |