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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 484 | . . . . 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 727 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑅 ∈ CRing) |
| 4 | ringlsmss1.2 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ⊆ 𝐵) | |
| 5 | 4 | sselda 3935 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐵) |
| 6 | 5 | adantlr 716 | . . . . . . . . 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 2737 | . . . . . . . . . . . . 13 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
| 10 | 8, 9 | lidlss 21182 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵) |
| 11 | 7, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
| 12 | 11 | sselda 3935 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐵) |
| 13 | 12 | adantr 480 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑖 ∈ 𝐵) |
| 14 | eqid 2737 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 15 | 8, 14 | crngcom 20201 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑒 ∈ 𝐵 ∧ 𝑖 ∈ 𝐵) → (𝑒(.r‘𝑅)𝑖) = (𝑖(.r‘𝑅)𝑒)) |
| 16 | 3, 6, 13, 15 | syl3anc 1374 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑒(.r‘𝑅)𝑖) = (𝑖(.r‘𝑅)𝑒)) |
| 17 | crngring 20195 | . . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 18 | 2, 17 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring) |
| 19 | 18 | ad2antrr 727 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑅 ∈ Ring) |
| 20 | 7 | ad2antrr 727 | . . . . . . . . 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 21195 | . . . . . . . . 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 2838 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
| 25 | 24 | adantllr 720 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
| 26 | 25 | adantr 480 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) ∧ 𝑎 = (𝑖(.r‘𝑅)𝑒)) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
| 27 | 1, 26 | eqeltrd 2837 | . . . 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 33490 | . . . . 5 ⊢ (𝜑 → (𝑎 ∈ (𝐼 × 𝐸) ↔ ∃𝑖 ∈ 𝐼 ∃𝑒 ∈ 𝐸 𝑎 = (𝑖(.r‘𝑅)𝑒))) |
| 31 | 30 | biimpa 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) → ∃𝑖 ∈ 𝐼 ∃𝑒 ∈ 𝐸 𝑎 = (𝑖(.r‘𝑅)𝑒)) |
| 32 | 27, 31 | r19.29vva 3198 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) → 𝑎 ∈ 𝐼) |
| 33 | 32 | ex 412 | . 2 ⊢ (𝜑 → (𝑎 ∈ (𝐼 × 𝐸) → 𝑎 ∈ 𝐼)) |
| 34 | 33 | ssrdv 3941 | 1 ⊢ (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 ⊆ wss 3903 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 .rcmulr 17190 LSSumclsm 19578 mulGrpcmgp 20090 Ringcrg 20183 CRingccrg 20184 LIdealclidl 21176 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-0g 17373 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-grp 18881 df-minusg 18882 df-sbg 18883 df-subg 19068 df-lsm 19580 df-cmn 19726 df-abl 19727 df-mgp 20091 df-rng 20103 df-ur 20132 df-ring 20185 df-cring 20186 df-subrg 20518 df-lmod 20828 df-lss 20898 df-sra 21140 df-rgmod 21141 df-lidl 21178 |
| This theorem is referenced by: idlsrgmulrss1 33608 |
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