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Mirrors > Home > ILE Home > Th. List > qus2idrng | GIF version |
Description: The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 14026 analog). (Contributed by AV, 23-Feb-2025.) |
Ref | Expression |
---|---|
qus2idrng.u | ⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆)) |
qus2idrng.i | ⊢ 𝐼 = (2Ideal‘𝑅) |
Ref | Expression |
---|---|
qus2idrng | ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qus2idrng.u | . . 3 ⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆)) | |
2 | 1 | a1i 9 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))) |
3 | eqidd 2194 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (Base‘𝑅) = (Base‘𝑅)) | |
4 | eqid 2193 | . 2 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
5 | eqid 2193 | . 2 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | simp3 1001 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅)) | |
7 | eqid 2193 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | eqid 2193 | . . . 4 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆) | |
9 | 7, 8 | eqger 13297 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → (𝑅 ~QG 𝑆) Er (Base‘𝑅)) |
10 | 6, 9 | syl 14 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (𝑅 ~QG 𝑆) Er (Base‘𝑅)) |
11 | rngabl 13434 | . . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | |
12 | 11 | 3ad2ant1 1020 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel) |
13 | ablnsg 13407 | . . . . 5 ⊢ (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅)) | |
14 | 12, 13 | syl 14 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (NrmSGrp‘𝑅) = (SubGrp‘𝑅)) |
15 | 6, 14 | eleqtrrd 2273 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (NrmSGrp‘𝑅)) |
16 | 7, 8, 4 | eqgcpbl 13301 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝑅) → ((𝑎(𝑅 ~QG 𝑆)𝑐 ∧ 𝑏(𝑅 ~QG 𝑆)𝑑) → (𝑎(+g‘𝑅)𝑏)(𝑅 ~QG 𝑆)(𝑐(+g‘𝑅)𝑑))) |
17 | 15, 16 | syl 14 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝑎(𝑅 ~QG 𝑆)𝑐 ∧ 𝑏(𝑅 ~QG 𝑆)𝑑) → (𝑎(+g‘𝑅)𝑏)(𝑅 ~QG 𝑆)(𝑐(+g‘𝑅)𝑑))) |
18 | qus2idrng.i | . . 3 ⊢ 𝐼 = (2Ideal‘𝑅) | |
19 | 7, 8, 18, 5 | 2idlcpblrng 14022 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝑎(𝑅 ~QG 𝑆)𝑐 ∧ 𝑏(𝑅 ~QG 𝑆)𝑑) → (𝑎(.r‘𝑅)𝑏)(𝑅 ~QG 𝑆)(𝑐(.r‘𝑅)𝑑))) |
20 | simp1 999 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Rng) | |
21 | 2, 3, 4, 5, 10, 17, 19, 20 | qusrng 13457 | 1 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 class class class wbr 4030 ‘cfv 5255 (class class class)co 5919 Er wer 6586 Basecbs 12621 +gcplusg 12698 .rcmulr 12699 /s cqus 12886 SubGrpcsubg 13240 NrmSGrpcnsg 13241 ~QG cqg 13242 Abelcabl 13358 Rngcrng 13431 2Idealc2idl 13998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-pre-ltirr 7986 ax-pre-lttrn 7988 ax-pre-ltadd 7990 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-tp 3627 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-tpos 6300 df-er 6589 df-ec 6591 df-qs 6595 df-pnf 8058 df-mnf 8059 df-ltxr 8061 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-ndx 12624 df-slot 12625 df-base 12627 df-sets 12628 df-iress 12629 df-plusg 12711 df-mulr 12712 df-sca 12714 df-vsca 12715 df-ip 12716 df-0g 12872 df-iimas 12888 df-qus 12889 df-mgm 12942 df-sgrp 12988 df-mnd 13001 df-grp 13078 df-minusg 13079 df-sbg 13080 df-subg 13243 df-nsg 13244 df-eqg 13245 df-cmn 13359 df-abl 13360 df-mgp 13420 df-rng 13432 df-oppr 13567 df-lssm 13852 df-sra 13934 df-rgmod 13935 df-lidl 13968 df-2idl 13999 |
This theorem is referenced by: (None) |
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