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Mirrors > Home > MPE Home > Th. List > mulgrhm2 | Structured version Visualization version GIF version |
Description: The powers of the element 1 give the unique ring homomorphism from ℤ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
Ref | Expression |
---|---|
mulgghm2.m | ⊢ · = (.g‘𝑅) |
mulgghm2.f | ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) |
mulgrhm.1 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
mulgrhm2 | ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) = {𝐹}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zringbas 20018 | . . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring) | |
2 | eqid 2752 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | rhmf 18920 | . . . . . . . . 9 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓:ℤ⟶(Base‘𝑅)) |
4 | 3 | adantl 473 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓:ℤ⟶(Base‘𝑅)) |
5 | 4 | feqmptd 6403 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 = (𝑛 ∈ ℤ ↦ (𝑓‘𝑛))) |
6 | rhmghm 18919 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓 ∈ (ℤring GrpHom 𝑅)) | |
7 | 6 | ad2antlr 765 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 𝑓 ∈ (ℤring GrpHom 𝑅)) |
8 | simpr 479 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
9 | 1zzd 11592 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 1 ∈ ℤ) | |
10 | eqid 2752 | . . . . . . . . . . 11 ⊢ (.g‘ℤring) = (.g‘ℤring) | |
11 | mulgghm2.m | . . . . . . . . . . 11 ⊢ · = (.g‘𝑅) | |
12 | 1, 10, 11 | ghmmulg 17865 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (ℤring GrpHom 𝑅) ∧ 𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑛 · (𝑓‘1))) |
13 | 7, 8, 9, 12 | syl3anc 1473 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑛 · (𝑓‘1))) |
14 | ax-1cn 10178 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ | |
15 | cnfldmulg 19972 | . . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℂ) → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) | |
16 | 14, 15 | mpan2 709 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) |
17 | 1z 11591 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ | |
18 | 16 | adantr 472 | . . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) |
19 | zringmulg 20020 | . . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℤring)1) = (𝑛 · 1)) | |
20 | 18, 19 | eqtr4d 2789 | . . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℂfld)1) = (𝑛(.g‘ℤring)1)) |
21 | 17, 20 | mpan2 709 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℂfld)1) = (𝑛(.g‘ℤring)1)) |
22 | zcn 11566 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
23 | 22 | mulid1d 10241 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛 · 1) = 𝑛) |
24 | 16, 21, 23 | 3eqtr3d 2794 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℤring)1) = 𝑛) |
25 | 24 | adantl 473 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑛(.g‘ℤring)1) = 𝑛) |
26 | 25 | fveq2d 6348 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑓‘𝑛)) |
27 | zring1 20023 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘ℤring) | |
28 | mulgrhm.1 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘𝑅) | |
29 | 27, 28 | rhm1 18924 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → (𝑓‘1) = 1 ) |
30 | 29 | ad2antlr 765 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘1) = 1 ) |
31 | 30 | oveq2d 6821 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑛 · (𝑓‘1)) = (𝑛 · 1 )) |
32 | 13, 26, 31 | 3eqtr3d 2794 | . . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘𝑛) = (𝑛 · 1 )) |
33 | 32 | mpteq2dva 4888 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → (𝑛 ∈ ℤ ↦ (𝑓‘𝑛)) = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) |
34 | 5, 33 | eqtrd 2786 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) |
35 | mulgghm2.f | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) | |
36 | 34, 35 | syl6eqr 2804 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 = 𝐹) |
37 | velsn 4329 | . . . . 5 ⊢ (𝑓 ∈ {𝐹} ↔ 𝑓 = 𝐹) | |
38 | 36, 37 | sylibr 224 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 ∈ {𝐹}) |
39 | 38 | ex 449 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓 ∈ {𝐹})) |
40 | 39 | ssrdv 3742 | . 2 ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) ⊆ {𝐹}) |
41 | 11, 35, 28 | mulgrhm 20040 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐹 ∈ (ℤring RingHom 𝑅)) |
42 | 41 | snssd 4477 | . 2 ⊢ (𝑅 ∈ Ring → {𝐹} ⊆ (ℤring RingHom 𝑅)) |
43 | 40, 42 | eqssd 3753 | 1 ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) = {𝐹}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1624 ∈ wcel 2131 {csn 4313 ↦ cmpt 4873 ⟶wf 6037 ‘cfv 6041 (class class class)co 6805 ℂcc 10118 1c1 10121 · cmul 10125 ℤcz 11561 Basecbs 16051 .gcmg 17733 GrpHom cghm 17850 1rcur 18693 Ringcrg 18739 RingHom crh 18906 ℂfldccnfld 19940 ℤringzring 20012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-inf2 8703 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-addf 10199 ax-mulf 10200 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-er 7903 df-map 8017 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-n0 11477 df-z 11562 df-dec 11678 df-uz 11872 df-fz 12512 df-seq 12988 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-starv 16150 df-tset 16154 df-ple 16155 df-ds 16158 df-unif 16159 df-0g 16296 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-mhm 17528 df-grp 17618 df-minusg 17619 df-mulg 17734 df-subg 17784 df-ghm 17851 df-cmn 18387 df-mgp 18682 df-ur 18694 df-ring 18741 df-cring 18742 df-rnghom 18909 df-subrg 18972 df-cnfld 19941 df-zring 20013 |
This theorem is referenced by: zrhval2 20051 zrhrhmb 20053 irinitoringc 42571 |
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