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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdsrspss | Structured version Visualization version GIF version |
Description: In a ring, an element 𝑋 divides 𝑌 iff the ideal generated by 𝑌 is a subset of the ideal generated by 𝑋 (Contributed by Thierry Arnoux, 22-Mar-2025.) |
Ref | Expression |
---|---|
dvdsrspss.b | ⊢ 𝐵 = (Base‘𝑅) |
dvdsrspss.k | ⊢ 𝐾 = (RSpan‘𝑅) |
dvdsrspss.d | ⊢ ∥ = (∥r‘𝑅) |
dvdsrspss.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
dvdsrspss.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
dvdsrspss.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
dvdsrspss | ⊢ (𝜑 → (𝑋 ∥ 𝑌 ↔ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsrspss.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | dvdsrspss.d | . . . 4 ⊢ ∥ = (∥r‘𝑅) | |
3 | eqid 2727 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 1, 2, 3 | dvdsr 20283 | . . 3 ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑡 ∈ 𝐵 (𝑡(.r‘𝑅)𝑋) = 𝑌)) |
5 | dvdsrspss.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 5 | biantrurd 532 | . . 3 ⊢ (𝜑 → (∃𝑡 ∈ 𝐵 (𝑡(.r‘𝑅)𝑋) = 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑡 ∈ 𝐵 (𝑡(.r‘𝑅)𝑋) = 𝑌))) |
7 | 4, 6 | bitr4id 290 | . 2 ⊢ (𝜑 → (𝑋 ∥ 𝑌 ↔ ∃𝑡 ∈ 𝐵 (𝑡(.r‘𝑅)𝑋) = 𝑌)) |
8 | dvdsrspss.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
9 | dvdsrspss.k | . . . . 5 ⊢ 𝐾 = (RSpan‘𝑅) | |
10 | 1, 3, 9 | rspsnel 33010 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ (𝐾‘{𝑋}) ↔ ∃𝑡 ∈ 𝐵 𝑌 = (𝑡(.r‘𝑅)𝑋))) |
11 | 8, 5, 10 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝐾‘{𝑋}) ↔ ∃𝑡 ∈ 𝐵 𝑌 = (𝑡(.r‘𝑅)𝑋))) |
12 | eqcom 2734 | . . . 4 ⊢ ((𝑡(.r‘𝑅)𝑋) = 𝑌 ↔ 𝑌 = (𝑡(.r‘𝑅)𝑋)) | |
13 | 12 | rexbii 3089 | . . 3 ⊢ (∃𝑡 ∈ 𝐵 (𝑡(.r‘𝑅)𝑋) = 𝑌 ↔ ∃𝑡 ∈ 𝐵 𝑌 = (𝑡(.r‘𝑅)𝑋)) |
14 | 11, 13 | bitr4di 289 | . 2 ⊢ (𝜑 → (𝑌 ∈ (𝐾‘{𝑋}) ↔ ∃𝑡 ∈ 𝐵 (𝑡(.r‘𝑅)𝑋) = 𝑌)) |
15 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐾‘{𝑋})) → 𝑅 ∈ Ring) |
16 | 5 | snssd 4808 | . . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝐵) |
17 | eqid 2727 | . . . . . . 7 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
18 | 9, 1, 17 | rspcl 21113 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → (𝐾‘{𝑋}) ∈ (LIdeal‘𝑅)) |
19 | 8, 16, 18 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (LIdeal‘𝑅)) |
20 | 19 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐾‘{𝑋})) → (𝐾‘{𝑋}) ∈ (LIdeal‘𝑅)) |
21 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐾‘{𝑋})) → 𝑌 ∈ (𝐾‘{𝑋})) | |
22 | 21 | snssd 4808 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐾‘{𝑋})) → {𝑌} ⊆ (𝐾‘{𝑋})) |
23 | 9, 17 | rspssp 21117 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐾‘{𝑋}) ∈ (LIdeal‘𝑅) ∧ {𝑌} ⊆ (𝐾‘{𝑋})) → (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) |
24 | 15, 20, 22, 23 | syl3anc 1369 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐾‘{𝑋})) → (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) |
25 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) → (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) | |
26 | dvdsrspss.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
27 | 26 | snssd 4808 | . . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝐵) |
28 | 9, 1 | rspssid 21114 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {𝑌} ⊆ 𝐵) → {𝑌} ⊆ (𝐾‘{𝑌})) |
29 | 8, 27, 28 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → {𝑌} ⊆ (𝐾‘{𝑌})) |
30 | snssg 4783 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (𝑌 ∈ (𝐾‘{𝑌}) ↔ {𝑌} ⊆ (𝐾‘{𝑌}))) | |
31 | 30 | biimpar 477 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐵 ∧ {𝑌} ⊆ (𝐾‘{𝑌})) → 𝑌 ∈ (𝐾‘{𝑌})) |
32 | 26, 29, 31 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝐾‘{𝑌})) |
33 | 32 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) → 𝑌 ∈ (𝐾‘{𝑌})) |
34 | 25, 33 | sseldd 3979 | . . 3 ⊢ ((𝜑 ∧ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) → 𝑌 ∈ (𝐾‘{𝑋})) |
35 | 24, 34 | impbida 800 | . 2 ⊢ (𝜑 → (𝑌 ∈ (𝐾‘{𝑋}) ↔ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋}))) |
36 | 7, 14, 35 | 3bitr2d 307 | 1 ⊢ (𝜑 → (𝑋 ∥ 𝑌 ↔ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 ⊆ wss 3944 {csn 4624 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 Basecbs 17165 .rcmulr 17219 Ringcrg 20157 ∥rcdsr 20275 LIdealclidl 21084 RSpancrsp 21085 |
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 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
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 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-0g 17408 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19062 df-mgp 20059 df-ur 20106 df-ring 20159 df-dvdsr 20278 df-subrg 20490 df-lmod 20727 df-lss 20798 df-lsp 20838 df-sra 21040 df-rgmod 21041 df-lidl 21086 df-rsp 21087 |
This theorem is referenced by: rspsnasso 33018 |
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