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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ssdifidl | Structured version Visualization version GIF version | ||
| Description: Let 𝑅 be a ring, and let 𝐼 be an ideal of 𝑅 disjoint with a set 𝑆. Then there exists an ideal 𝑖, maximal among the set 𝑃 of ideals containing 𝐼 and disjoint with 𝑆. (Contributed by Thierry Arnoux, 3-Jun-2025.) | 
| Ref | Expression | 
|---|---|
| ssdifidl.1 | ⊢ 𝐵 = (Base‘𝑅) | 
| ssdifidl.2 | ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| ssdifidl.3 | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | 
| ssdifidl.4 | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | 
| ssdifidl.5 | ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅) | 
| ssdifidl.6 | ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)} | 
| Ref | Expression | 
|---|---|
| ssdifidl | ⊢ (𝜑 → ∃𝑖 ∈ 𝑃 ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ineq2 4213 | . . . . . . 7 ⊢ (𝑝 = 𝐼 → (𝑆 ∩ 𝑝) = (𝑆 ∩ 𝐼)) | |
| 2 | 1 | eqeq1d 2738 | . . . . . 6 ⊢ (𝑝 = 𝐼 → ((𝑆 ∩ 𝑝) = ∅ ↔ (𝑆 ∩ 𝐼) = ∅)) | 
| 3 | sseq2 4009 | . . . . . 6 ⊢ (𝑝 = 𝐼 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ 𝐼)) | |
| 4 | 2, 3 | anbi12d 632 | . . . . 5 ⊢ (𝑝 = 𝐼 → (((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝) ↔ ((𝑆 ∩ 𝐼) = ∅ ∧ 𝐼 ⊆ 𝐼))) | 
| 5 | ssdifidl.3 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | |
| 6 | ssdifidl.5 | . . . . . 6 ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅) | |
| 7 | ssidd 4006 | . . . . . 6 ⊢ (𝜑 → 𝐼 ⊆ 𝐼) | |
| 8 | 6, 7 | jca 511 | . . . . 5 ⊢ (𝜑 → ((𝑆 ∩ 𝐼) = ∅ ∧ 𝐼 ⊆ 𝐼)) | 
| 9 | 4, 5, 8 | elrabd 3693 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}) | 
| 10 | ssdifidl.6 | . . . 4 ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)} | |
| 11 | 9, 10 | eleqtrrdi 2851 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑃) | 
| 12 | 11 | ne0d 4341 | . 2 ⊢ (𝜑 → 𝑃 ≠ ∅) | 
| 13 | ssdifidl.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 14 | ssdifidl.2 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 15 | 14 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧)) → 𝑅 ∈ Ring) | 
| 16 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧)) → 𝐼 ∈ (LIdeal‘𝑅)) | 
| 17 | ssdifidl.4 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
| 18 | 17 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧)) → 𝑆 ⊆ 𝐵) | 
| 19 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧)) → (𝑆 ∩ 𝐼) = ∅) | 
| 20 | simpr1 1194 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧)) → 𝑧 ⊆ 𝑃) | |
| 21 | simpr2 1195 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧)) → 𝑧 ≠ ∅) | |
| 22 | simpr3 1196 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧)) → [⊊] Or 𝑧) | |
| 23 | 13, 15, 16, 18, 19, 10, 20, 21, 22 | ssdifidllem 33485 | . . . 4 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧)) → ∪ 𝑧 ∈ 𝑃) | 
| 24 | 23 | ex 412 | . . 3 ⊢ (𝜑 → ((𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝑃)) | 
| 25 | 24 | alrimiv 1926 | . 2 ⊢ (𝜑 → ∀𝑧((𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝑃)) | 
| 26 | fvex 6918 | . . . 4 ⊢ (LIdeal‘𝑅) ∈ V | |
| 27 | 10, 26 | rabex2 5340 | . . 3 ⊢ 𝑃 ∈ V | 
| 28 | 27 | zornn0 10549 | . 2 ⊢ ((𝑃 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝑃 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝑃)) → ∃𝑖 ∈ 𝑃 ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗) | 
| 29 | 12, 25, 28 | syl2anc 584 | 1 ⊢ (𝜑 → ∃𝑖 ∈ 𝑃 ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wal 1537 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3435 ∩ cin 3949 ⊆ wss 3950 ⊊ wpss 3951 ∅c0 4332 ∪ cuni 4906 Or wor 5590 ‘cfv 6560 [⊊] crpss 7743 Basecbs 17248 Ringcrg 20231 LIdealclidl 21217 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-ac2 10504 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 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 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-rpss 7744 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-ac 10157 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-subrg 20571 df-lmod 20861 df-lss 20931 df-sra 21173 df-rgmod 21174 df-lidl 21219 | 
| This theorem is referenced by: ssdifidlprm 33487 | 
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