Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringlsmss1 | Structured version Visualization version GIF version |
Description: The product of an ideal 𝐼 of a commutative ring 𝑅 with some set E is a subset of the ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.) |
Ref | Expression |
---|---|
ringlsmss.1 | ⊢ 𝐵 = (Base‘𝑅) |
ringlsmss.2 | ⊢ 𝐺 = (mulGrp‘𝑅) |
ringlsmss.3 | ⊢ × = (LSSum‘𝐺) |
ringlsmss1.1 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
ringlsmss1.2 | ⊢ (𝜑 → 𝐸 ⊆ 𝐵) |
ringlsmss1.3 | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
Ref | Expression |
---|---|
ringlsmss1 | ⊢ (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) ∧ 𝑎 = (𝑖(.r‘𝑅)𝑒)) → 𝑎 = (𝑖(.r‘𝑅)𝑒)) | |
2 | ringlsmss1.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
3 | 2 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑅 ∈ CRing) |
4 | ringlsmss1.2 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ⊆ 𝐵) | |
5 | 4 | sselda 3891 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐵) |
6 | 5 | adantlr 715 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐵) |
7 | ringlsmss1.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | |
8 | ringlsmss.1 | . . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑅) | |
9 | eqid 2734 | . . . . . . . . . . . . 13 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
10 | 8, 9 | lidlss 20220 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵) |
11 | 7, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
12 | 11 | sselda 3891 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐵) |
13 | 12 | adantr 484 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑖 ∈ 𝐵) |
14 | eqid 2734 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
15 | 8, 14 | crngcom 19552 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑒 ∈ 𝐵 ∧ 𝑖 ∈ 𝐵) → (𝑒(.r‘𝑅)𝑖) = (𝑖(.r‘𝑅)𝑒)) |
16 | 3, 6, 13, 15 | syl3anc 1373 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑒(.r‘𝑅)𝑖) = (𝑖(.r‘𝑅)𝑒)) |
17 | crngring 19546 | . . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
18 | 2, 17 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | 18 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑅 ∈ Ring) |
20 | 7 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝐼 ∈ (LIdeal‘𝑅)) |
21 | simplr 769 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑖 ∈ 𝐼) | |
22 | 9, 8, 14 | lidlmcl 20227 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼)) → (𝑒(.r‘𝑅)𝑖) ∈ 𝐼) |
23 | 19, 20, 6, 21, 22 | syl22anc 839 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑒(.r‘𝑅)𝑖) ∈ 𝐼) |
24 | 16, 23 | eqeltrrd 2835 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
25 | 24 | adantllr 719 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
26 | 25 | adantr 484 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) ∧ 𝑎 = (𝑖(.r‘𝑅)𝑒)) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
27 | 1, 26 | eqeltrd 2834 | . . . 4 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) ∧ 𝑎 = (𝑖(.r‘𝑅)𝑒)) → 𝑎 ∈ 𝐼) |
28 | ringlsmss.2 | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅) | |
29 | ringlsmss.3 | . . . . . 6 ⊢ × = (LSSum‘𝐺) | |
30 | 8, 14, 28, 29, 11, 4 | elringlsm 31267 | . . . . 5 ⊢ (𝜑 → (𝑎 ∈ (𝐼 × 𝐸) ↔ ∃𝑖 ∈ 𝐼 ∃𝑒 ∈ 𝐸 𝑎 = (𝑖(.r‘𝑅)𝑒))) |
31 | 30 | biimpa 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) → ∃𝑖 ∈ 𝐼 ∃𝑒 ∈ 𝐸 𝑎 = (𝑖(.r‘𝑅)𝑒)) |
32 | 27, 31 | r19.29vva 3245 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) → 𝑎 ∈ 𝐼) |
33 | 32 | ex 416 | . 2 ⊢ (𝜑 → (𝑎 ∈ (𝐼 × 𝐸) → 𝑎 ∈ 𝐼)) |
34 | 33 | ssrdv 3897 | 1 ⊢ (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 ⊆ wss 3857 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 .rcmulr 16768 LSSumclsm 18995 mulGrpcmgp 19476 Ringcrg 19534 CRingccrg 19535 LIdealclidl 20179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-lsm 18997 df-cmn 19144 df-mgp 19477 df-ur 19489 df-ring 19536 df-cring 19537 df-subrg 19770 df-lmod 19873 df-lss 19941 df-sra 20181 df-rgmod 20182 df-lidl 20183 |
This theorem is referenced by: idlsrgmulrss1 31342 |
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