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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 21481 | . . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring) | |
2 | eqid 2734 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | rhmf 20501 | . . . . . . . . 9 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓:ℤ⟶(Base‘𝑅)) |
4 | 3 | adantl 481 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓:ℤ⟶(Base‘𝑅)) |
5 | 4 | feqmptd 6976 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 = (𝑛 ∈ ℤ ↦ (𝑓‘𝑛))) |
6 | rhmghm 20500 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓 ∈ (ℤring GrpHom 𝑅)) | |
7 | 6 | ad2antlr 727 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 𝑓 ∈ (ℤring GrpHom 𝑅)) |
8 | simpr 484 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
9 | 1zzd 12645 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → 1 ∈ ℤ) | |
10 | eqid 2734 | . . . . . . . . . . 11 ⊢ (.g‘ℤring) = (.g‘ℤring) | |
11 | mulgghm2.m | . . . . . . . . . . 11 ⊢ · = (.g‘𝑅) | |
12 | 1, 10, 11 | ghmmulg 19258 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (ℤring GrpHom 𝑅) ∧ 𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑛 · (𝑓‘1))) |
13 | 7, 8, 9, 12 | syl3anc 1370 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑛 · (𝑓‘1))) |
14 | ax-1cn 11210 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ | |
15 | cnfldmulg 21433 | . . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℂ) → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) | |
16 | 14, 15 | mpan2 691 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) |
17 | 1z 12644 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ | |
18 | 16 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℂfld)1) = (𝑛 · 1)) |
19 | zringmulg 21484 | . . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℤring)1) = (𝑛 · 1)) | |
20 | 18, 19 | eqtr4d 2777 | . . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑛(.g‘ℂfld)1) = (𝑛(.g‘ℤring)1)) |
21 | 17, 20 | mpan2 691 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℂfld)1) = (𝑛(.g‘ℤring)1)) |
22 | zcn 12615 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
23 | 22 | mulridd 11275 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (𝑛 · 1) = 𝑛) |
24 | 16, 21, 23 | 3eqtr3d 2782 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℤring)1) = 𝑛) |
25 | 24 | adantl 481 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑛(.g‘ℤring)1) = 𝑛) |
26 | 25 | fveq2d 6910 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘(𝑛(.g‘ℤring)1)) = (𝑓‘𝑛)) |
27 | zring1 21487 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘ℤring) | |
28 | mulgrhm.1 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘𝑅) | |
29 | 27, 28 | rhm1 20505 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤring RingHom 𝑅) → (𝑓‘1) = 1 ) |
30 | 29 | ad2antlr 727 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘1) = 1 ) |
31 | 30 | oveq2d 7446 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑛 · (𝑓‘1)) = (𝑛 · 1 )) |
32 | 13, 26, 31 | 3eqtr3d 2782 | . . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) ∧ 𝑛 ∈ ℤ) → (𝑓‘𝑛) = (𝑛 · 1 )) |
33 | 32 | mpteq2dva 5247 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → (𝑛 ∈ ℤ ↦ (𝑓‘𝑛)) = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) |
34 | 5, 33 | eqtrd 2774 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) |
35 | mulgghm2.f | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) | |
36 | 34, 35 | eqtr4di 2792 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 = 𝐹) |
37 | velsn 4646 | . . . . 5 ⊢ (𝑓 ∈ {𝐹} ↔ 𝑓 = 𝐹) | |
38 | 36, 37 | sylibr 234 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ (ℤring RingHom 𝑅)) → 𝑓 ∈ {𝐹}) |
39 | 38 | ex 412 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑓 ∈ (ℤring RingHom 𝑅) → 𝑓 ∈ {𝐹})) |
40 | 39 | ssrdv 4000 | . 2 ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) ⊆ {𝐹}) |
41 | 11, 35, 28 | mulgrhm 21505 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐹 ∈ (ℤring RingHom 𝑅)) |
42 | 41 | snssd 4813 | . 2 ⊢ (𝑅 ∈ Ring → {𝐹} ⊆ (ℤring RingHom 𝑅)) |
43 | 40, 42 | eqssd 4012 | 1 ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) = {𝐹}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {csn 4630 ↦ cmpt 5230 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 1c1 11153 · cmul 11157 ℤcz 12610 Basecbs 17244 .gcmg 19097 GrpHom cghm 19242 1rcur 20198 Ringcrg 20250 RingHom crh 20485 ℂfldccnfld 21381 ℤringczring 21474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-addf 11231 ax-mulf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-seq 14039 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-grp 18966 df-minusg 18967 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-rhm 20488 df-subrng 20562 df-subrg 20586 df-cnfld 21382 df-zring 21475 |
This theorem is referenced by: irinitoringc 21507 zrhval2 21536 zrhrhmb 21538 |
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