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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringlsmss2 | Structured version Visualization version GIF version |
Description: The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
Ref | Expression |
---|---|
ringlsmss.1 | ⊢ 𝐵 = (Base‘𝑅) |
ringlsmss.2 | ⊢ 𝐺 = (mulGrp‘𝑅) |
ringlsmss.3 | ⊢ × = (LSSum‘𝐺) |
ringlsmss2.1 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringlsmss2.2 | ⊢ (𝜑 → 𝐸 ⊆ 𝐵) |
ringlsmss2.3 | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
Ref | Expression |
---|---|
ringlsmss2 | ⊢ (𝜑 → (𝐸 × 𝐼) ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 489 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐸 × 𝐼)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) ∧ 𝑎 = (𝑒(.r‘𝑅)𝑖)) → 𝑎 = (𝑒(.r‘𝑅)𝑖)) | |
2 | ringlsmss2.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
3 | 2 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ Ring) |
4 | ringlsmss2.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | |
5 | 4 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ (LIdeal‘𝑅)) |
6 | ringlsmss2.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ⊆ 𝐵) | |
7 | 6 | sselda 3888 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐵) |
8 | 7 | adantr 485 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) → 𝑒 ∈ 𝐵) |
9 | simpr 489 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼) | |
10 | eqid 2759 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
11 | ringlsmss.1 | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅) | |
12 | eqid 2759 | . . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
13 | 10, 11, 12 | lidlmcl 20043 | . . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼)) → (𝑒(.r‘𝑅)𝑖) ∈ 𝐼) |
14 | 3, 5, 8, 9, 13 | syl22anc 838 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) → (𝑒(.r‘𝑅)𝑖) ∈ 𝐼) |
15 | 14 | adantllr 719 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐸 × 𝐼)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) → (𝑒(.r‘𝑅)𝑖) ∈ 𝐼) |
16 | 15 | adantr 485 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐸 × 𝐼)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) ∧ 𝑎 = (𝑒(.r‘𝑅)𝑖)) → (𝑒(.r‘𝑅)𝑖) ∈ 𝐼) |
17 | 1, 16 | eqeltrd 2851 | . . . 4 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐸 × 𝐼)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑖 ∈ 𝐼) ∧ 𝑎 = (𝑒(.r‘𝑅)𝑖)) → 𝑎 ∈ 𝐼) |
18 | ringlsmss.2 | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅) | |
19 | ringlsmss.3 | . . . . . 6 ⊢ × = (LSSum‘𝐺) | |
20 | 11, 10 | lidlss 20036 | . . . . . . 7 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵) |
21 | 4, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
22 | 11, 12, 18, 19, 6, 21 | elringlsm 31087 | . . . . 5 ⊢ (𝜑 → (𝑎 ∈ (𝐸 × 𝐼) ↔ ∃𝑒 ∈ 𝐸 ∃𝑖 ∈ 𝐼 𝑎 = (𝑒(.r‘𝑅)𝑖))) |
23 | 22 | biimpa 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐸 × 𝐼)) → ∃𝑒 ∈ 𝐸 ∃𝑖 ∈ 𝐼 𝑎 = (𝑒(.r‘𝑅)𝑖)) |
24 | 17, 23 | r19.29vva 3255 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐸 × 𝐼)) → 𝑎 ∈ 𝐼) |
25 | 24 | ex 417 | . 2 ⊢ (𝜑 → (𝑎 ∈ (𝐸 × 𝐼) → 𝑎 ∈ 𝐼)) |
26 | 25 | ssrdv 3894 | 1 ⊢ (𝜑 → (𝐸 × 𝐼) ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∃wrex 3069 ⊆ wss 3854 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 .rcmulr 16609 LSSumclsm 18811 mulGrpcmgp 19292 Ringcrg 19350 LIdealclidl 19995 |
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-rmo 3076 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-ress 16534 df-plusg 16621 df-mulr 16622 df-sca 16624 df-vsca 16625 df-ip 16626 df-0g 16758 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-grp 18157 df-minusg 18158 df-sbg 18159 df-subg 18328 df-lsm 18813 df-mgp 19293 df-ur 19305 df-ring 19352 df-subrg 19586 df-lmod 19689 df-lss 19757 df-sra 19997 df-rgmod 19998 df-lidl 19999 |
This theorem is referenced by: idlsrgmulrss2 31163 |
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