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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pidlnz | Structured version Visualization version GIF version |
Description: A principal ideal generated by a nonzero element is not the zero ideal. (Contributed by Thierry Arnoux, 11-Apr-2024.) |
Ref | Expression |
---|---|
pidlnz.1 | ⊢ 𝐵 = (Base‘𝑅) |
pidlnz.2 | ⊢ 0 = (0g‘𝑅) |
pidlnz.3 | ⊢ 𝐾 = (RSpan‘𝑅) |
Ref | Expression |
---|---|
pidlnz | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐾‘{𝑋}) ≠ { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑅 ∈ Ring) | |
2 | simpl2 1193 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 ∈ 𝐵) | |
3 | pidlnz.1 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
4 | pidlnz.3 | . . . . . . 7 ⊢ 𝐾 = (RSpan‘𝑅) | |
5 | 3, 4 | rspsnid 33386 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (𝐾‘{𝑋})) |
6 | 1, 2, 5 | syl2anc 584 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 ∈ (𝐾‘{𝑋})) |
7 | simpr 484 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → (𝐾‘{𝑋}) = { 0 }) | |
8 | 6, 7 | eleqtrd 2842 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 ∈ { 0 }) |
9 | elsni 4641 | . . . 4 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 ) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 = 0 ) |
11 | simpl3 1194 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 ≠ 0 ) | |
12 | 11 | neneqd 2944 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → ¬ 𝑋 = 0 ) |
13 | 10, 12 | pm2.65da 817 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ¬ (𝐾‘{𝑋}) = { 0 }) |
14 | 13 | neqned 2946 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐾‘{𝑋}) ≠ { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 {csn 4624 ‘cfv 6559 Basecbs 17243 0gc0g 17480 Ringcrg 20226 RSpancrsp 21209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-sca 17309 df-vsca 17310 df-ip 17311 df-0g 17482 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-grp 18950 df-subg 19137 df-mgp 20134 df-ur 20175 df-ring 20228 df-subrg 20562 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21164 df-rgmod 21165 df-rsp 21211 |
This theorem is referenced by: pidlnzb 33437 drngidl 33448 minplym1p 33737 |
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