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Mirrors > Home > MPE Home > Th. List > 2idlcpbl | Structured version Visualization version GIF version |
Description: The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.) |
Ref | Expression |
---|---|
2idlcpblrng.x | ⊢ 𝑋 = (Base‘𝑅) |
2idlcpblrng.r | ⊢ 𝐸 = (𝑅 ~QG 𝑆) |
2idlcpblrng.i | ⊢ 𝐼 = (2Ideal‘𝑅) |
2idlcpblrng.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
2idlcpbl | ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringrng 20233 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Rng) | |
2 | 1 | adantr 479 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Rng) |
3 | simpr 483 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ 𝐼) | |
4 | eqid 2725 | . . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
5 | eqid 2725 | . . . . 5 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
6 | eqid 2725 | . . . . 5 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
7 | 2idlcpblrng.i | . . . . 5 ⊢ 𝐼 = (2Ideal‘𝑅) | |
8 | 4, 5, 6, 7 | 2idlelb 21160 | . . . 4 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))) |
9 | 8 | simplbi 496 | . . 3 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅)) |
10 | 4 | lidlsubg 21131 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅)) |
11 | 9, 10 | sylan2 591 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅)) |
12 | 2idlcpblrng.x | . . 3 ⊢ 𝑋 = (Base‘𝑅) | |
13 | 2idlcpblrng.r | . . 3 ⊢ 𝐸 = (𝑅 ~QG 𝑆) | |
14 | 2idlcpblrng.t | . . 3 ⊢ · = (.r‘𝑅) | |
15 | 12, 13, 7, 14 | 2idlcpblrng 21178 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷))) |
16 | 2, 3, 11, 15 | syl3anc 1368 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 .rcmulr 17237 SubGrpcsubg 19083 ~QG cqg 19085 Rngcrng 20104 Ringcrg 20185 opprcoppr 20284 LIdealclidl 21114 2Idealc2idl 21156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-eqg 19088 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-subrg 20520 df-lmod 20757 df-lss 20828 df-sra 21070 df-rgmod 21071 df-lidl 21116 df-2idl 21157 |
This theorem is referenced by: qus1 21181 qusrhm 21183 qusmul2idl 21186 qsidomlem1 33264 qsidomlem2 33265 |
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