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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 33071 | . . . . 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 476 | . . . 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 480 | . 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 7413 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π Β· π) = (πΌ Β· π½)) | |
| 11 | 10 | eleq1d 2812 | . . . . 5 β’ ((π = πΌ β§ π = π½) β ((π Β· π) β π β (πΌ Β· π½) β π)) |
| 12 | simpl 482 | . . . . . . 7 β’ ((π = πΌ β§ π = π½) β π = πΌ) | |
| 13 | 12 | eleq1d 2812 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π β π β πΌ β π)) |
| 14 | simpr 484 | . . . . . . 7 β’ ((π = πΌ β§ π = π½) β π = π½) | |
| 15 | 14 | eleq1d 2812 | . . . . . 6 β’ ((π = πΌ β§ π = π½) β (π β π β π½ β π)) |
| 16 | 13, 15 | orbi12d 915 | . . . . 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 3616 | . . 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 417 | . 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 837 | 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 395 β¨ wo 844 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 βcfv 6536 (class class class)co 7404 Basecbs 17150 .rcmulr 17204 CRingccrg 20136 LIdealclidl 21062 PrmIdealcprmidl 33058 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-subrg 20468 df-lmod 20705 df-lss 20776 df-lsp 20816 df-sra 21018 df-rgmod 21019 df-lidl 21064 df-rsp 21065 df-prmidl 33059 |
| This theorem is referenced by: rhmpreimaprmidl 33075 |
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