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Mathbox for Alexander van der Vekens |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rnglidl1 | Structured version Visualization version GIF version | ||
| Description: The base set of every non-unital ring is an ideal. For unital rings, such ideals are called "unit ideals", see lidl1 20837. (Contributed by AV, 19-Feb-2025.) |
| Ref | Expression |
|---|---|
| rnglidl0.u | β’ π = (LIdealβπ ) |
| rnglidl1.b | β’ π΅ = (Baseβπ ) |
| Ref | Expression |
|---|---|
| rnglidl1 | β’ (π β Rng β π΅ β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rnglidl1.b | . . . 4 β’ π΅ = (Baseβπ ) | |
| 2 | 1 | eqimssi 4041 | . . 3 β’ π΅ β (Baseβπ ) |
| 3 | 2 | a1i 11 | . 2 β’ (π β Rng β π΅ β (Baseβπ )) |
| 4 | rnggrp 46640 | . . 3 β’ (π β Rng β π β Grp) | |
| 5 | 1 | grpbn0 18847 | . . 3 β’ (π β Grp β π΅ β β ) |
| 6 | 4, 5 | syl 17 | . 2 β’ (π β Rng β π΅ β β ) |
| 7 | eqid 2732 | . . . 4 β’ (+gβπ ) = (+gβπ ) | |
| 8 | 4 | adantr 481 | . . . 4 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π β Grp) |
| 9 | simpl 483 | . . . . 5 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π β Rng) | |
| 10 | 1 | eqcomi 2741 | . . . . . . . . 9 β’ (Baseβπ ) = π΅ |
| 11 | 10 | eleq2i 2825 | . . . . . . . 8 β’ (π₯ β (Baseβπ ) β π₯ β π΅) |
| 12 | 11 | biimpi 215 | . . . . . . 7 β’ (π₯ β (Baseβπ ) β π₯ β π΅) |
| 13 | 12 | 3ad2ant1 1133 | . . . . . 6 β’ ((π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅) β π₯ β π΅) |
| 14 | 13 | adantl 482 | . . . . 5 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π₯ β π΅) |
| 15 | simpr2 1195 | . . . . 5 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π¦ β π΅) | |
| 16 | eqid 2732 | . . . . . 6 β’ (.rβπ ) = (.rβπ ) | |
| 17 | 1, 16 | rngcl 46649 | . . . . 5 β’ ((π β Rng β§ π₯ β π΅ β§ π¦ β π΅) β (π₯(.rβπ )π¦) β π΅) |
| 18 | 9, 14, 15, 17 | syl3anc 1371 | . . . 4 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β (π₯(.rβπ )π¦) β π΅) |
| 19 | simpr3 1196 | . . . 4 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π§ β π΅) | |
| 20 | 1, 7, 8, 18, 19 | grpcld 18829 | . . 3 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β ((π₯(.rβπ )π¦)(+gβπ )π§) β π΅) |
| 21 | 20 | ralrimivvva 3203 | . 2 β’ (π β Rng β βπ₯ β (Baseβπ )βπ¦ β π΅ βπ§ β π΅ ((π₯(.rβπ )π¦)(+gβπ )π§) β π΅) |
| 22 | rnglidl0.u | . . 3 β’ π = (LIdealβπ ) | |
| 23 | eqid 2732 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
| 24 | 22, 23, 7, 16 | islidl 20826 | . 2 β’ (π΅ β π β (π΅ β (Baseβπ ) β§ π΅ β β β§ βπ₯ β (Baseβπ )βπ¦ β π΅ βπ§ β π΅ ((π₯(.rβπ )π¦)(+gβπ )π§) β π΅)) |
| 25 | 3, 6, 21, 24 | syl3anbrc 1343 | 1 β’ (π β Rng β π΅ β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 β wss 3947 β c0 4321 βcfv 6540 (class class class)co 7405 Basecbs 17140 +gcplusg 17193 .rcmulr 17194 Grpcgrp 18815 LIdealclidl 20775 Rngcrng 46634 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-sca 17209 df-vsca 17210 df-ip 17211 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-abl 19645 df-mgp 19982 df-lss 20535 df-sra 20777 df-rgmod 20778 df-lidl 20779 df-rng 46635 |
| This theorem is referenced by: (None) |
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