Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lidlmmgm | Structured version Visualization version GIF version |
Description: The multiplicative group of a (left) ideal of a ring is a magma. (Contributed by AV, 17-Feb-2020.) |
Ref | Expression |
---|---|
lidlabl.l | ⊢ 𝐿 = (LIdeal‘𝑅) |
lidlabl.i | ⊢ 𝐼 = (𝑅 ↾s 𝑈) |
Ref | Expression |
---|---|
lidlmmgm | ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (mulGrp‘𝐼) ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlabl.l | . . . . . . . 8 ⊢ 𝐿 = (LIdeal‘𝑅) | |
2 | lidlabl.i | . . . . . . . 8 ⊢ 𝐼 = (𝑅 ↾s 𝑈) | |
3 | 1, 2 | lidlbas 45481 | . . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈) |
4 | eleq1a 2834 | . . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) = 𝑈 → (Base‘𝐼) ∈ 𝐿)) | |
5 | 3, 4 | mpd 15 | . . . . . 6 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿) |
6 | 5 | anim2i 617 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (𝑅 ∈ Ring ∧ (Base‘𝐼) ∈ 𝐿)) |
7 | 6 | adantr 481 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑅 ∈ Ring ∧ (Base‘𝐼) ∈ 𝐿)) |
8 | 1, 2 | lidlssbas 45480 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅)) |
9 | 8 | adantl 482 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (Base‘𝐼) ⊆ (Base‘𝑅)) |
10 | 9 | sseld 3920 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅))) |
11 | 10 | com12 32 | . . . . . 6 ⊢ (𝑎 ∈ (Base‘𝐼) → ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝑎 ∈ (Base‘𝑅))) |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼)) → ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝑎 ∈ (Base‘𝑅))) |
13 | 12 | impcom 408 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → 𝑎 ∈ (Base‘𝑅)) |
14 | simprr 770 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → 𝑏 ∈ (Base‘𝐼)) | |
15 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | eqid 2738 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
17 | 1, 15, 16 | lidlmcl 20488 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ (Base‘𝐼) ∈ 𝐿) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼)) |
18 | 7, 13, 14, 17 | syl12anc 834 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼)) |
19 | 18 | ralrimivva 3123 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼)) |
20 | fvex 6787 | . . . 4 ⊢ (mulGrp‘𝐼) ∈ V | |
21 | eqid 2738 | . . . . . 6 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼) | |
22 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝐼) = (Base‘𝐼) | |
23 | 21, 22 | mgpbas 19726 | . . . . 5 ⊢ (Base‘𝐼) = (Base‘(mulGrp‘𝐼)) |
24 | eqid 2738 | . . . . . 6 ⊢ (.r‘𝐼) = (.r‘𝐼) | |
25 | 21, 24 | mgpplusg 19724 | . . . . 5 ⊢ (.r‘𝐼) = (+g‘(mulGrp‘𝐼)) |
26 | 23, 25 | ismgm 18327 | . . . 4 ⊢ ((mulGrp‘𝐼) ∈ V → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼))) |
27 | 20, 26 | mp1i 13 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼))) |
28 | 2, 16 | ressmulr 17017 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼)) |
29 | 28 | eqcomd 2744 | . . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝐼) = (.r‘𝑅)) |
30 | 29 | adantl 482 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (.r‘𝐼) = (.r‘𝑅)) |
31 | 30 | oveqdr 7303 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏)) |
32 | 31 | eleq1d 2823 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → ((𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))) |
33 | 32 | 2ralbidva 3128 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))) |
34 | 27, 33 | bitrd 278 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))) |
35 | 19, 34 | mpbird 256 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (mulGrp‘𝐼) ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 ↾s cress 16941 .rcmulr 16963 Mgmcmgm 18324 mulGrpcmgp 19720 Ringcrg 19783 LIdealclidl 20432 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-mgp 19721 df-ur 19738 df-ring 19785 df-subrg 20022 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-lidl 20436 |
This theorem is referenced by: lidlmsgrp 45484 |
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