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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlmulssprm | Structured version Visualization version GIF version |
Description: Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 ⊆ 𝑃 or 𝐽 ⊆ 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.) |
Ref | Expression |
---|---|
idlmulssprm.1 | ⊢ × = (LSSum‘(mulGrp‘𝑅)) |
idlmulssprm.2 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
idlmulssprm.3 | ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅)) |
idlmulssprm.4 | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
idlmulssprm.5 | ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) |
idlmulssprm.6 | ⊢ (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃) |
Ref | Expression |
---|---|
idlmulssprm | ⊢ (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlmulssprm.2 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | idlmulssprm.3 | . 2 ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅)) | |
3 | idlmulssprm.4 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | |
4 | idlmulssprm.5 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) | |
5 | 3, 4 | jca 516 | . 2 ⊢ (𝜑 → (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) |
6 | idlmulssprm.6 | . . . . . 6 ⊢ (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃) | |
7 | 6 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝐼 × 𝐽) ⊆ 𝑃) |
8 | eqid 2759 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | eqid 2759 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | eqid 2759 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
11 | idlmulssprm.1 | . . . . . 6 ⊢ × = (LSSum‘(mulGrp‘𝑅)) | |
12 | eqid 2759 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
13 | 8, 12 | lidlss 20036 | . . . . . . . 8 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ (Base‘𝑅)) |
14 | 3, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑅)) |
15 | 14 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝐼 ⊆ (Base‘𝑅)) |
16 | 8, 12 | lidlss 20036 | . . . . . . . 8 ⊢ (𝐽 ∈ (LIdeal‘𝑅) → 𝐽 ⊆ (Base‘𝑅)) |
17 | 4, 16 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑅)) |
18 | 17 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝐽 ⊆ (Base‘𝑅)) |
19 | simplr 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑥 ∈ 𝐼) | |
20 | simpr 489 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽) | |
21 | 8, 9, 10, 11, 15, 18, 19, 20 | elringlsmd 31088 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑥(.r‘𝑅)𝑦) ∈ (𝐼 × 𝐽)) |
22 | 7, 21 | sseldd 3889 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑃) |
23 | 22 | anasss 471 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑃) |
24 | 23 | ralrimivva 3118 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥(.r‘𝑅)𝑦) ∈ 𝑃) |
25 | 8, 9 | prmidl 31121 | . 2 ⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥(.r‘𝑅)𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)) |
26 | 1, 2, 5, 24, 25 | syl1111anc 839 | 1 ⊢ (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∨ wo 845 = wceq 1539 ∈ wcel 2112 ∀wral 3068 ⊆ wss 3854 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 .rcmulr 16609 LSSumclsm 18811 mulGrpcmgp 19292 Ringcrg 19350 LIdealclidl 19995 PrmIdealcprmidl 31116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-plusg 16621 df-sca 16624 df-vsca 16625 df-ip 16626 df-lsm 18813 df-mgp 19293 df-lss 19757 df-sra 19997 df-rgmod 19998 df-lidl 19999 df-prmidl 31117 |
This theorem is referenced by: zarclsun 31326 |
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