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 1187 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑅 ∈ Ring) | |
2 | simpl2 1188 | . . . . . 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 30942 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (𝐾‘{𝑋})) |
6 | 1, 2, 5 | syl2anc 586 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 ∈ (𝐾‘{𝑋})) |
7 | simpr 487 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → (𝐾‘{𝑋}) = { 0 }) | |
8 | 6, 7 | eleqtrd 2913 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 ∈ { 0 }) |
9 | elsni 4558 | . . . 4 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 ) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 = 0 ) |
11 | simpl3 1189 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → 𝑋 ≠ 0 ) | |
12 | 11 | neneqd 3011 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝐾‘{𝑋}) = { 0 }) → ¬ 𝑋 = 0 ) |
13 | 10, 12 | pm2.65da 815 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ¬ (𝐾‘{𝑋}) = { 0 }) |
14 | 13 | neqned 3013 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐾‘{𝑋}) ≠ { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3006 {csn 4541 ‘cfv 6329 Basecbs 16459 0gc0g 16689 Ringcrg 19273 RSpancrsp 19916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5240 ax-pr 5304 ax-un 7437 ax-cnex 10569 ax-resscn 10570 ax-1cn 10571 ax-icn 10572 ax-addcl 10573 ax-addrcl 10574 ax-mulcl 10575 ax-mulrcl 10576 ax-mulcom 10577 ax-addass 10578 ax-mulass 10579 ax-distr 10580 ax-i2m1 10581 ax-1ne0 10582 ax-1rid 10583 ax-rnegex 10584 ax-rrecex 10585 ax-cnre 10586 ax-pre-lttri 10587 ax-pre-lttrn 10588 ax-pre-ltadd 10589 ax-pre-mulgt0 10590 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3007 df-nel 3111 df-ral 3130 df-rex 3131 df-reu 3132 df-rmo 3133 df-rab 3134 df-v 3475 df-sbc 3752 df-csb 3860 df-dif 3915 df-un 3917 df-in 3919 df-ss 3928 df-pss 3930 df-nul 4268 df-if 4442 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4813 df-int 4851 df-iun 4895 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5434 df-eprel 5439 df-po 5448 df-so 5449 df-fr 5488 df-we 5490 df-xp 5535 df-rel 5536 df-cnv 5537 df-co 5538 df-dm 5539 df-rn 5540 df-res 5541 df-ima 5542 df-pred 6122 df-ord 6168 df-on 6169 df-lim 6170 df-suc 6171 df-iota 6288 df-fun 6331 df-fn 6332 df-f 6333 df-f1 6334 df-fo 6335 df-f1o 6336 df-fv 6337 df-riota 7089 df-ov 7134 df-oprab 7135 df-mpo 7136 df-om 7557 df-wrecs 7923 df-recs 7984 df-rdg 8022 df-er 8265 df-en 8486 df-dom 8487 df-sdom 8488 df-pnf 10653 df-mnf 10654 df-xr 10655 df-ltxr 10656 df-le 10657 df-sub 10848 df-neg 10849 df-nn 11615 df-2 11677 df-3 11678 df-4 11679 df-5 11680 df-6 11681 df-7 11682 df-8 11683 df-ndx 16462 df-slot 16463 df-base 16465 df-sets 16466 df-ress 16467 df-plusg 16554 df-mulr 16555 df-sca 16557 df-vsca 16558 df-ip 16559 df-0g 16691 df-mgm 17828 df-sgrp 17877 df-mnd 17888 df-grp 18082 df-subg 18252 df-mgp 19216 df-ur 19228 df-ring 19275 df-subrg 19506 df-lmod 19609 df-lss 19677 df-lsp 19717 df-sra 19917 df-rgmod 19918 df-rsp 19920 |
This theorem is referenced by: (None) |
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