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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > drnglidl1ne0 | Structured version Visualization version GIF version |
Description: In a nonzero ring, the zero ideal is different of the unit ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
Ref | Expression |
---|---|
drnglidl1ne0.1 | ⊢ 0 = (0g‘𝑅) |
drnglidl1ne0.2 | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
drnglidl1ne0 | ⊢ (𝑅 ∈ NzRing → 𝐵 ≠ { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nzrring 20454 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
2 | drnglidl1ne0.2 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | eqid 2725 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 20201 | . . 3 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ∈ 𝐵) |
6 | drnglidl1ne0.1 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
7 | 3, 6 | nzrnz 20453 | . . 3 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 ) |
8 | nelsn 4665 | . . 3 ⊢ ((1r‘𝑅) ≠ 0 → ¬ (1r‘𝑅) ∈ { 0 }) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝑅 ∈ NzRing → ¬ (1r‘𝑅) ∈ { 0 }) |
10 | nelne1 3029 | . 2 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ ¬ (1r‘𝑅) ∈ { 0 }) → 𝐵 ≠ { 0 }) | |
11 | 5, 9, 10 | syl2anc 582 | 1 ⊢ (𝑅 ∈ NzRing → 𝐵 ≠ { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 {csn 4625 ‘cfv 6543 Basecbs 17174 0gc0g 17415 1rcur 20120 Ringcrg 20172 NzRingcnzr 20450 |
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 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 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 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mgp 20074 df-ur 20121 df-ring 20174 df-nzr 20451 |
This theorem is referenced by: drngmxidl 33235 |
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