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Mirrors > Home > MPE Home > Th. List > chrrhm | Structured version Visualization version GIF version |
Description: The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
chrrhm | ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmrcl1 19082 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring) | |
2 | eqid 2825 | . . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
3 | 2 | zrhrhm 20227 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅)) |
4 | 1, 3 | syl 17 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅)) |
5 | zringbas 20191 | . . . . . . 7 ⊢ ℤ = (Base‘ℤring) | |
6 | eqid 2825 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 5, 6 | rhmf 19089 | . . . . . 6 ⊢ ((ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅) → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅)) |
8 | ffn 6282 | . . . . . 6 ⊢ ((ℤRHom‘𝑅):ℤ⟶(Base‘𝑅) → (ℤRHom‘𝑅) Fn ℤ) | |
9 | 4, 7, 8 | 3syl 18 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (ℤRHom‘𝑅) Fn ℤ) |
10 | eqid 2825 | . . . . . . 7 ⊢ (chr‘𝑅) = (chr‘𝑅) | |
11 | 10 | chrcl 20241 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (chr‘𝑅) ∈ ℕ0) |
12 | nn0z 11735 | . . . . . 6 ⊢ ((chr‘𝑅) ∈ ℕ0 → (chr‘𝑅) ∈ ℤ) | |
13 | 1, 11, 12 | 3syl 18 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑅) ∈ ℤ) |
14 | fvco2 6524 | . . . . 5 ⊢ (((ℤRHom‘𝑅) Fn ℤ ∧ (chr‘𝑅) ∈ ℤ) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅)))) | |
15 | 9, 13, 14 | syl2anc 579 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅)))) |
16 | eqid 2825 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | 10, 2, 16 | chrid 20242 | . . . . . 6 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘(chr‘𝑅)) = (0g‘𝑅)) |
18 | 1, 17 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((ℤRHom‘𝑅)‘(chr‘𝑅)) = (0g‘𝑅)) |
19 | 18 | fveq2d 6441 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))) = (𝐹‘(0g‘𝑅))) |
20 | 15, 19 | eqtrd 2861 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘(0g‘𝑅))) |
21 | rhmco 19100 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅)) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤring RingHom 𝑆)) | |
22 | 4, 21 | mpdan 678 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤring RingHom 𝑆)) |
23 | rhmrcl2 19083 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring) | |
24 | eqid 2825 | . . . . . . 7 ⊢ (ℤRHom‘𝑆) = (ℤRHom‘𝑆) | |
25 | 24 | zrhrhmb 20226 | . . . . . 6 ⊢ (𝑆 ∈ Ring → ((𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤring RingHom 𝑆) ↔ (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆))) |
26 | 23, 25 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤring RingHom 𝑆) ↔ (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆))) |
27 | 22, 26 | mpbid 224 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆)) |
28 | 27 | fveq1d 6439 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = ((ℤRHom‘𝑆)‘(chr‘𝑅))) |
29 | rhmghm 19088 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
30 | eqid 2825 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
31 | 16, 30 | ghmid 18024 | . . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) |
32 | 29, 31 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) |
33 | 20, 28, 32 | 3eqtr3d 2869 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0g‘𝑆)) |
34 | eqid 2825 | . . . 4 ⊢ (chr‘𝑆) = (chr‘𝑆) | |
35 | 34, 24, 30 | chrdvds 20243 | . . 3 ⊢ ((𝑆 ∈ Ring ∧ (chr‘𝑅) ∈ ℤ) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0g‘𝑆))) |
36 | 23, 13, 35 | syl2anc 579 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0g‘𝑆))) |
37 | 33, 36 | mpbird 249 | 1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ∘ ccom 5350 Fn wfn 6122 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 ℕ0cn0 11625 ℤcz 11711 ∥ cdvds 15364 Basecbs 16229 0gc0g 16460 GrpHom cghm 18015 Ringcrg 18908 RingHom crh 19075 ℤringzring 20185 ℤRHomczrh 20215 chrcchr 20217 |
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-inf2 8822 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 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
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-rmo 3125 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-int 4700 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-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-fz 12627 df-fl 12895 df-mod 12971 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-dvds 15365 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-od 18306 df-cmn 18555 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-rnghom 19078 df-subrg 19141 df-cnfld 20114 df-zring 20186 df-zrh 20219 df-chr 20221 |
This theorem is referenced by: (None) |
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