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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pidlnzb | Structured version Visualization version GIF version |
Description: A principal ideal is nonzero iff it is generated by a nonzero elements (Contributed by Thierry Arnoux, 22-Mar-2025.) |
Ref | Expression |
---|---|
pidlnzb.1 | ⊢ 𝐵 = (Base‘𝑅) |
pidlnzb.2 | ⊢ 0 = (0g‘𝑅) |
pidlnzb.3 | ⊢ 𝐾 = (RSpan‘𝑅) |
Ref | Expression |
---|---|
pidlnzb | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 ↔ (𝐾‘{𝑋}) ≠ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pidlnzb.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | pidlnzb.2 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
3 | pidlnzb.3 | . . . 4 ⊢ 𝐾 = (RSpan‘𝑅) | |
4 | 1, 2, 3 | pidlnz 33027 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐾‘{𝑋}) ≠ { 0 }) |
5 | 4 | 3expa 1116 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≠ 0 ) → (𝐾‘{𝑋}) ≠ { 0 }) |
6 | sneq 4634 | . . . . . . . 8 ⊢ (𝑋 = 0 → {𝑋} = { 0 }) | |
7 | 6 | fveq2d 6895 | . . . . . . 7 ⊢ (𝑋 = 0 → (𝐾‘{𝑋}) = (𝐾‘{ 0 })) |
8 | 7 | adantl 481 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 = 0 ) → (𝐾‘{𝑋}) = (𝐾‘{ 0 })) |
9 | 3, 2 | rsp0 21123 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 }) |
10 | 9 | ad2antrr 725 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 = 0 ) → (𝐾‘{ 0 }) = { 0 }) |
11 | 8, 10 | eqtrd 2767 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 = 0 ) → (𝐾‘{𝑋}) = { 0 }) |
12 | 11 | ex 412 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 = 0 → (𝐾‘{𝑋}) = { 0 })) |
13 | 12 | necon3d 2956 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝐾‘{𝑋}) ≠ { 0 } → 𝑋 ≠ 0 )) |
14 | 13 | imp 406 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝐾‘{𝑋}) ≠ { 0 }) → 𝑋 ≠ 0 ) |
15 | 5, 14 | impbida 800 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 ↔ (𝐾‘{𝑋}) ≠ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 {csn 4624 ‘cfv 6542 Basecbs 17171 0gc0g 17412 Ringcrg 20164 RSpancrsp 21092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-subg 19069 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-subrg 20497 df-lmod 20734 df-lss 20805 df-lsp 20845 df-sra 21047 df-rgmod 21048 df-rsp 21094 |
This theorem is referenced by: irngnminplynz 33318 |
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