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Mirrors > Home > MPE Home > Th. List > lpi0 | Structured version Visualization version GIF version |
Description: The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
lpival.p | β’ π = (LPIdealβπ ) |
lpi0.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
lpi0 | β’ (π β Ring β { 0 } β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
2 | lpi0.z | . . . 4 β’ 0 = (0gβπ ) | |
3 | 1, 2 | ring0cl 20196 | . . 3 β’ (π β Ring β 0 β (Baseβπ )) |
4 | eqid 2728 | . . . . 5 β’ (RSpanβπ ) = (RSpanβπ ) | |
5 | 4, 2 | rsp0 21127 | . . . 4 β’ (π β Ring β ((RSpanβπ )β{ 0 }) = { 0 }) |
6 | 5 | eqcomd 2734 | . . 3 β’ (π β Ring β { 0 } = ((RSpanβπ )β{ 0 })) |
7 | sneq 4634 | . . . . 5 β’ (π = 0 β {π} = { 0 }) | |
8 | 7 | fveq2d 6895 | . . . 4 β’ (π = 0 β ((RSpanβπ )β{π}) = ((RSpanβπ )β{ 0 })) |
9 | 8 | rspceeqv 3630 | . . 3 β’ (( 0 β (Baseβπ ) β§ { 0 } = ((RSpanβπ )β{ 0 })) β βπ β (Baseβπ ){ 0 } = ((RSpanβπ )β{π})) |
10 | 3, 6, 9 | syl2anc 583 | . 2 β’ (π β Ring β βπ β (Baseβπ ){ 0 } = ((RSpanβπ )β{π})) |
11 | lpival.p | . . 3 β’ π = (LPIdealβπ ) | |
12 | 11, 4, 1 | islpidl 21208 | . 2 β’ (π β Ring β ({ 0 } β π β βπ β (Baseβπ ){ 0 } = ((RSpanβπ )β{π}))) |
13 | 10, 12 | mpbird 257 | 1 β’ (π β Ring β { 0 } β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βwrex 3066 {csn 4624 βcfv 6542 Basecbs 17173 0gc0g 17414 Ringcrg 20166 RSpancrsp 21096 LPIdealclpidl 21203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-subg 19071 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-subrg 20501 df-lmod 20738 df-lss 20809 df-lsp 20849 df-sra 21051 df-rgmod 21052 df-rsp 21098 df-lpidl 21205 |
This theorem is referenced by: drnglpir 21215 zringlpir 21386 |
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