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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 21092. (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 4037 | . . 3 β’ π΅ β (Baseβπ ) |
| 3 | 2 | a1i 11 | . 2 β’ (π β Rng β π΅ β (Baseβπ )) |
| 4 | rnggrp 20063 | . . 3 β’ (π β Rng β π β Grp) | |
| 5 | 1 | grpbn0 18896 | . . 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 2726 | . . . 4 β’ (+gβπ ) = (+gβπ ) | |
| 8 | 4 | adantr 480 | . . . 4 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π β Grp) |
| 9 | simpl 482 | . . . . 5 β’ ((π β Rng β§ (π₯ β (Baseβπ ) β§ π¦ β π΅ β§ π§ β π΅)) β π β Rng) | |
| 10 | 1 | eqcomi 2735 | . . . . . . . . 9 β’ (Baseβπ ) = π΅ |
| 11 | 10 | eleq2i 2819 | . . . . . . . 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 481 | . . . . 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 2726 | . . . . . 6 β’ (.rβπ ) = (.rβπ ) | |
| 17 | 1, 16 | rngcl 20069 | . . . . 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 18877 | . . 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 3197 | . 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 2726 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
| 24 | 22, 23, 7, 16 | islidl 21074 | . 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 β wss 3943 β c0 4317 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 .rcmulr 17207 Grpcgrp 18863 Rngcrng 20057 LIdealclidl 21065 |
| 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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| 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-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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-sca 17222 df-vsca 17223 df-ip 17224 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-abl 19703 df-mgp 20040 df-rng 20058 df-lss 20779 df-sra 21021 df-rgmod 21022 df-lidl 21067 |
| This theorem is referenced by: lidl1 21092 |
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