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| Mirrors > Home > MPE Home > Th. List > 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 21143. (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 4042 | . . 3 β’ π΅ β (Baseβπ ) |
| 3 | 2 | a1i 11 | . 2 β’ (π β Rng β π΅ β (Baseβπ )) |
| 4 | rnggrp 20112 | . . 3 β’ (π β Rng β π β Grp) | |
| 5 | 1 | grpbn0 18937 | . . 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 2728 | . . . 4 β’ (+gβπ ) = (+gβπ ) | |
| 8 | 4 | adantr 479 | . . . 4 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π β Grp) |
| 9 | simpl 481 | . . . . 5 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π β Rng) | |
| 10 | 1 | eqcomi 2737 | . . . . . . . . 9 β’ (Baseβπ ) = π΅ |
| 11 | 10 | eleq2i 2821 | . . . . . . . 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 1130 | . . . . . 6 β’ ((π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅) β π₯ β π΅) |
| 14 | 13 | adantl 480 | . . . . 5 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π₯ β π΅) |
| 15 | simpr2 1192 | . . . . 5 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π¦ β π΅) | |
| 16 | eqid 2728 | . . . . . 6 β’ (.rβπ ) = (.rβπ ) | |
| 17 | 1, 16 | rngcl 20118 | . . . . 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 1368 | . . . 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 1193 | . . . 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 18918 | . . 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 3201 | . 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 2728 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
| 24 | 22, 23, 7, 16 | islidl 21125 | . 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 1340 | 1 β’ (π β Rng β π΅ β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 βwral 3058 β wss 3949 β c0 4326 βcfv 6553 (class class class)co 7426 Basecbs 17189 +gcplusg 17242 .rcmulr 17243 Grpcgrp 18904 Rngcrng 20106 LIdealclidl 21116 |
| 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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-sca 17258 df-vsca 17259 df-ip 17260 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-abl 19752 df-mgp 20089 df-rng 20107 df-lss 20830 df-sra 21072 df-rgmod 21073 df-lidl 21118 |
| This theorem is referenced by: lidl1 21143 |
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