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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 2735 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 1, 2, 3 | dvdsr 20379 | . . 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 | elrspsn 21268 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ (𝐾‘{𝑋}) ↔ ∃𝑡 ∈ 𝐵 𝑌 = (𝑡(.r‘𝑅)𝑋))) |
11 | 8, 5, 10 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝐾‘{𝑋}) ↔ ∃𝑡 ∈ 𝐵 𝑌 = (𝑡(.r‘𝑅)𝑋))) |
12 | eqcom 2742 | . . . 4 ⊢ ((𝑡(.r‘𝑅)𝑋) = 𝑌 ↔ 𝑌 = (𝑡(.r‘𝑅)𝑋)) | |
13 | 12 | rexbii 3092 | . . 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 4814 | . . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝐵) |
17 | eqid 2735 | . . . . . . 7 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
18 | 9, 1, 17 | rspcl 21263 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → (𝐾‘{𝑋}) ∈ (LIdeal‘𝑅)) |
19 | 8, 16, 18 | syl2anc 584 | . . . . 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 4814 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐾‘{𝑋})) → {𝑌} ⊆ (𝐾‘{𝑋})) |
23 | 9, 17 | rspssp 21267 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐾‘{𝑋}) ∈ (LIdeal‘𝑅) ∧ {𝑌} ⊆ (𝐾‘{𝑋})) → (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) |
24 | 15, 20, 22, 23 | syl3anc 1370 | . . 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 4814 | . . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝐵) |
28 | 9, 1 | rspssid 21264 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {𝑌} ⊆ 𝐵) → {𝑌} ⊆ (𝐾‘{𝑌})) |
29 | 8, 27, 28 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → {𝑌} ⊆ (𝐾‘{𝑌})) |
30 | snssg 4788 | . . . . . . 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 584 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝐾‘{𝑌})) |
33 | 32 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) → 𝑌 ∈ (𝐾‘{𝑌})) |
34 | 25, 33 | sseldd 3996 | . . 3 ⊢ ((𝜑 ∧ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})) → 𝑌 ∈ (𝐾‘{𝑋})) |
35 | 24, 34 | impbida 801 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 Ringcrg 20251 ∥rcdsr 20371 LIdealclidl 21234 RSpancrsp 21235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-mgp 20153 df-ur 20200 df-ring 20253 df-dvdsr 20374 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 |
This theorem is referenced by: rspsnasso 33396 |
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