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Mirrors > Home > MPE Home > Th. List > ringlghm | Structured version Visualization version GIF version |
Description: Left-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 |
---|---|
ringlghm | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 GrpHom 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringlghm.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2778 | . 2 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
3 | ringgrp 19028 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
4 | 3 | adantr 473 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp) |
5 | ringlghm.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
6 | 1, 5 | ringcl 19037 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑥) ∈ 𝐵) |
7 | 6 | 3expa 1098 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑥) ∈ 𝐵) |
8 | 7 | fmpttd 6704 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)):𝐵⟶𝐵) |
9 | 3anass 1076 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) | |
10 | 1, 2, 5 | ringdi 19042 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑋 · (𝑦(+g‘𝑅)𝑧)) = ((𝑋 · 𝑦)(+g‘𝑅)(𝑋 · 𝑧))) |
11 | 9, 10 | sylan2br 585 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑋 · (𝑦(+g‘𝑅)𝑧)) = ((𝑋 · 𝑦)(+g‘𝑅)(𝑋 · 𝑧))) |
12 | 11 | anassrs 460 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑋 · (𝑦(+g‘𝑅)𝑧)) = ((𝑋 · 𝑦)(+g‘𝑅)(𝑋 · 𝑧))) |
13 | 1, 2 | ringacl 19054 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑅)𝑧) ∈ 𝐵) |
14 | 13 | 3expb 1100 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(+g‘𝑅)𝑧) ∈ 𝐵) |
15 | 14 | adantlr 702 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(+g‘𝑅)𝑧) ∈ 𝐵) |
16 | oveq2 6986 | . . . . 5 ⊢ (𝑥 = (𝑦(+g‘𝑅)𝑧) → (𝑋 · 𝑥) = (𝑋 · (𝑦(+g‘𝑅)𝑧))) | |
17 | eqid 2778 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)) | |
18 | ovex 7010 | . . . . 5 ⊢ (𝑋 · (𝑦(+g‘𝑅)𝑧)) ∈ V | |
19 | 16, 17, 18 | fvmpt 6597 | . . . 4 ⊢ ((𝑦(+g‘𝑅)𝑧) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥))‘(𝑦(+g‘𝑅)𝑧)) = (𝑋 · (𝑦(+g‘𝑅)𝑧))) |
20 | 15, 19 | syl 17 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥))‘(𝑦(+g‘𝑅)𝑧)) = (𝑋 · (𝑦(+g‘𝑅)𝑧))) |
21 | oveq2 6986 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑋 · 𝑥) = (𝑋 · 𝑦)) | |
22 | ovex 7010 | . . . . . 6 ⊢ (𝑋 · 𝑦) ∈ V | |
23 | 21, 17, 22 | fvmpt 6597 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥))‘𝑦) = (𝑋 · 𝑦)) |
24 | oveq2 6986 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑋 · 𝑥) = (𝑋 · 𝑧)) | |
25 | ovex 7010 | . . . . . 6 ⊢ (𝑋 · 𝑧) ∈ V | |
26 | 24, 17, 25 | fvmpt 6597 | . . . . 5 ⊢ (𝑧 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥))‘𝑧) = (𝑋 · 𝑧)) |
27 | 23, 26 | oveqan12d 6997 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (((𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥))‘𝑦)(+g‘𝑅)((𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥))‘𝑧)) = ((𝑋 · 𝑦)(+g‘𝑅)(𝑋 · 𝑧))) |
28 | 27 | adantl 474 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (((𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥))‘𝑦)(+g‘𝑅)((𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥))‘𝑧)) = ((𝑋 · 𝑦)(+g‘𝑅)(𝑋 · 𝑧))) |
29 | 12, 20, 28 | 3eqtr4d 2824 | . 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))) |
30 | 1, 1, 2, 2, 4, 4, 8, 29 | isghmd 18141 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 GrpHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ↦ cmpt 5009 ‘cfv 6190 (class class class)co 6978 Basecbs 16342 +gcplusg 16424 .rcmulr 16425 Grpcgrp 17894 GrpHom cghm 18129 Ringcrg 19023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-plusg 16437 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-grp 17897 df-ghm 18130 df-mgp 18966 df-ring 19025 |
This theorem is referenced by: gsummulc2 19083 |
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