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Mirrors > Home > MPE Home > Th. List > ringrghm | Structured version Visualization version GIF version |
Description: Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.) |
Ref | Expression |
---|---|
ringlghm.b | ⊢ 𝐵 = (Base‘𝑅) |
ringlghm.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
ringrghm | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 GrpHom 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringlghm.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2825 | . 2 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
3 | ringgrp 18913 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
4 | 3 | adantr 474 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp) |
5 | ringlghm.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
6 | 1, 5 | ringcl 18922 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑥 · 𝑋) ∈ 𝐵) |
7 | 6 | 3expa 1151 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) → (𝑥 · 𝑋) ∈ 𝐵) |
8 | 7 | an32s 642 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 · 𝑋) ∈ 𝐵) |
9 | 8 | fmpttd 6639 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)):𝐵⟶𝐵) |
10 | df-3an 1113 | . . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵)) | |
11 | 1, 2, 5 | ringdir 18928 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑦(+g‘𝑅)𝑧) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)(𝑧 · 𝑋))) |
12 | 10, 11 | sylan2br 588 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵)) → ((𝑦(+g‘𝑅)𝑧) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)(𝑧 · 𝑋))) |
13 | 12 | anass1rs 645 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦(+g‘𝑅)𝑧) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)(𝑧 · 𝑋))) |
14 | 1, 2 | ringacl 18939 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑅)𝑧) ∈ 𝐵) |
15 | 14 | 3expb 1153 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(+g‘𝑅)𝑧) ∈ 𝐵) |
16 | 15 | adantlr 706 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(+g‘𝑅)𝑧) ∈ 𝐵) |
17 | oveq1 6917 | . . . . 5 ⊢ (𝑥 = (𝑦(+g‘𝑅)𝑧) → (𝑥 · 𝑋) = ((𝑦(+g‘𝑅)𝑧) · 𝑋)) | |
18 | eqid 2825 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)) = (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)) | |
19 | ovex 6942 | . . . . 5 ⊢ ((𝑦(+g‘𝑅)𝑧) · 𝑋) ∈ V | |
20 | 17, 18, 19 | fvmpt 6533 | . . . 4 ⊢ ((𝑦(+g‘𝑅)𝑧) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘(𝑦(+g‘𝑅)𝑧)) = ((𝑦(+g‘𝑅)𝑧) · 𝑋)) |
21 | 16, 20 | syl 17 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘(𝑦(+g‘𝑅)𝑧)) = ((𝑦(+g‘𝑅)𝑧) · 𝑋)) |
22 | oveq1 6917 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋)) | |
23 | ovex 6942 | . . . . . 6 ⊢ (𝑦 · 𝑋) ∈ V | |
24 | 22, 18, 23 | fvmpt 6533 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘𝑦) = (𝑦 · 𝑋)) |
25 | oveq1 6917 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 · 𝑋) = (𝑧 · 𝑋)) | |
26 | ovex 6942 | . . . . . 6 ⊢ (𝑧 · 𝑋) ∈ V | |
27 | 25, 18, 26 | fvmpt 6533 | . . . . 5 ⊢ (𝑧 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘𝑧) = (𝑧 · 𝑋)) |
28 | 24, 27 | oveqan12d 6929 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘𝑦)(+g‘𝑅)((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘𝑧)) = ((𝑦 · 𝑋)(+g‘𝑅)(𝑧 · 𝑋))) |
29 | 28 | adantl 475 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘𝑦)(+g‘𝑅)((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘𝑧)) = ((𝑦 · 𝑋)(+g‘𝑅)(𝑧 · 𝑋))) |
30 | 13, 21, 29 | 3eqtr4d 2871 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘(𝑦(+g‘𝑅)𝑧)) = (((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘𝑦)(+g‘𝑅)((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋))‘𝑧))) |
31 | 1, 1, 2, 2, 4, 4, 9, 30 | isghmd 18027 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 GrpHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ↦ cmpt 4954 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 .rcmulr 16313 Grpcgrp 17783 GrpHom cghm 18015 Ringcrg 18908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-plusg 16325 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-ghm 18016 df-mgp 18851 df-ring 18910 |
This theorem is referenced by: gsummulc1 18967 fidomndrnglem 19674 |
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