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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 19824 | . . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring) | |
2 | eqid 2622 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | rhmf 18726 | . . . . . . . . 9 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓:ℤ⟶(Base‘𝑅)) |
4 | 3 | adantl 482 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓:ℤ⟶(Base‘𝑅)) |
5 | 4 | feqmptd 6249 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 = (𝑛 ∈ ℤ ↦ (𝑓‘𝑛))) |
6 | rhmghm 18725 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓 ∈ (ℤring GrpHom 𝑅)) | |
7 | 6 | ad2antlr 763 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 𝑓 ∈ (ℤring GrpHom 𝑅)) |
8 | simpr 477 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
9 | 1zzd 11408 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 1 ∈ ℤ) | |
10 | eqid 2622 | . . . . . . . . . . 11 ⊢ (.g‘ℤring) = (.g‘ℤring) | |
11 | mulgghm2.m | . . . . . . . . . . 11 ⊢ · = (.g‘𝑅) | |
12 | 1, 10, 11 | ghmmulg 17672 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (ℤring GrpHom 𝑅) ∧ 𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑛 · (𝑓‘1))) |
13 | 7, 8, 9, 12 | syl3anc 1326 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑛 · (𝑓‘1))) |
14 | ax-1cn 9994 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ | |
15 | cnfldmulg 19778 | . . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℂ) → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) | |
16 | 14, 15 | mpan2 707 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) |
17 | 1z 11407 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ | |
18 | 16 | adantr 481 | . . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) |
19 | zringmulg 19826 | . . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℤring)1) = (𝑛 · 1)) | |
20 | 18, 19 | eqtr4d 2659 | . . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℂfld)1) = (𝑛(.g‘ℤring)1)) |
21 | 17, 20 | mpan2 707 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℂfld)1) = (𝑛(.g‘ℤring)1)) |
22 | zcn 11382 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
23 | 22 | mulid1d 10057 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛 · 1) = 𝑛) |
24 | 16, 21, 23 | 3eqtr3d 2664 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℤring)1) = 𝑛) |
25 | 24 | adantl 482 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑛(.g‘ℤring)1) = 𝑛) |
26 | 25 | fveq2d 6195 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑓‘𝑛)) |
27 | zring1 19829 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘ℤring) | |
28 | mulgrhm.1 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘𝑅) | |
29 | 27, 28 | rhm1 18730 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → (𝑓‘1) = 1 ) |
30 | 29 | ad2antlr 763 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘1) = 1 ) |
31 | 30 | oveq2d 6666 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑛 · (𝑓‘1)) = (𝑛 · 1 )) |
32 | 13, 26, 31 | 3eqtr3d 2664 | . . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘𝑛) = (𝑛 · 1 )) |
33 | 32 | mpteq2dva 4744 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → (𝑛 ∈ ℤ ↦ (𝑓‘𝑛)) = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) |
34 | 5, 33 | eqtrd 2656 | . . . . . 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 2674 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 = 𝐹) |
37 | velsn 4193 | . . . . 5 ⊢ (𝑓 ∈ {𝐹} ↔ 𝑓 = 𝐹) | |
38 | 36, 37 | sylibr 224 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 ∈ {𝐹}) |
39 | 38 | ex 450 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓 ∈ {𝐹})) |
40 | 39 | ssrdv 3609 | . 2 ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) ⊆ {𝐹}) |
41 | 11, 35, 28 | mulgrhm 19846 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐹 ∈ (ℤring RingHom 𝑅)) |
42 | 41 | snssd 4340 | . 2 ⊢ (𝑅 ∈ Ring → {𝐹} ⊆ (ℤring RingHom 𝑅)) |
43 | 40, 42 | eqssd 3620 | 1 ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) = {𝐹}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 1c1 9937 · cmul 9941 ℤcz 11377 Basecbs 15857 .gcmg 17540 GrpHom cghm 17657 1rcur 18501 Ringcrg 18547 RingHom crh 18712 ℂfldccnfld 19746 ℤringzring 19818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-cnfld 19747 df-zring 19819 |
This theorem is referenced by: zrhval2 19857 zrhrhmb 19859 irinitoringc 42069 |
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