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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pidlnz | Structured version Visualization version GIF version | ||
| Description: A principal ideal generated by a nonzero element is not the zero ideal. (Contributed by Thierry Arnoux, 11-Apr-2024.) |
| Ref | Expression |
|---|---|
| pidlnz.1 | β’ π΅ = (Baseβπ ) |
| pidlnz.2 | β’ 0 = (0gβπ ) |
| pidlnz.3 | β’ πΎ = (RSpanβπ ) |
| Ref | Expression |
|---|---|
| pidlnz | β’ ((π β Ring β§ π β π΅ β§ π β 0 ) β (πΎβ{π}) β { 0 }) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl1 1188 | . . . . . 6 β’ (((π β Ring β§ π β π΅ β§ π β 0 ) β§ (πΎβ{π}) = { 0 }) β π β Ring) | |
| 2 | simpl2 1189 | . . . . . 6 β’ (((π β Ring β§ π β π΅ β§ π β 0 ) β§ (πΎβ{π}) = { 0 }) β π β π΅) | |
| 3 | pidlnz.1 | . . . . . . 7 β’ π΅ = (Baseβπ ) | |
| 4 | pidlnz.3 | . . . . . . 7 β’ πΎ = (RSpanβπ ) | |
| 5 | 3, 4 | rspsnid 32991 | . . . . . 6 β’ ((π β Ring β§ π β π΅) β π β (πΎβ{π})) |
| 6 | 1, 2, 5 | syl2anc 583 | . . . . 5 β’ (((π β Ring β§ π β π΅ β§ π β 0 ) β§ (πΎβ{π}) = { 0 }) β π β (πΎβ{π})) |
| 7 | simpr 484 | . . . . 5 β’ (((π β Ring β§ π β π΅ β§ π β 0 ) β§ (πΎβ{π}) = { 0 }) β (πΎβ{π}) = { 0 }) | |
| 8 | 6, 7 | eleqtrd 2829 | . . . 4 β’ (((π β Ring β§ π β π΅ β§ π β 0 ) β§ (πΎβ{π}) = { 0 }) β π β { 0 }) |
| 9 | elsni 4640 | . . . 4 β’ (π β { 0 } β π = 0 ) | |
| 10 | 8, 9 | syl 17 | . . 3 β’ (((π β Ring β§ π β π΅ β§ π β 0 ) β§ (πΎβ{π}) = { 0 }) β π = 0 ) |
| 11 | simpl3 1190 | . . . 4 β’ (((π β Ring β§ π β π΅ β§ π β 0 ) β§ (πΎβ{π}) = { 0 }) β π β 0 ) | |
| 12 | 11 | neneqd 2939 | . . 3 β’ (((π β Ring β§ π β π΅ β§ π β 0 ) β§ (πΎβ{π}) = { 0 }) β Β¬ π = 0 ) |
| 13 | 10, 12 | pm2.65da 814 | . 2 β’ ((π β Ring β§ π β π΅ β§ π β 0 ) β Β¬ (πΎβ{π}) = { 0 }) |
| 14 | 13 | neqned 2941 | 1 β’ ((π β Ring β§ π β π΅ β§ π β 0 ) β (πΎβ{π}) β { 0 }) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 {csn 4623 βcfv 6536 Basecbs 17151 0gc0g 17392 Ringcrg 20136 RSpancrsp 21064 |
| 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-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 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-subg 19048 df-mgp 20038 df-ur 20085 df-ring 20138 df-subrg 20469 df-lmod 20706 df-lss 20777 df-lsp 20817 df-sra 21019 df-rgmod 21020 df-rsp 21066 |
| This theorem is referenced by: pidlnzb 33046 drngidl 33057 minplym1p 33292 |
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