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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 18542 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring) | |
2 | eqid 2610 | . . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
3 | 2 | zrhrhm 19679 | . . . . . . 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 19643 | . . . . . . 7 ⊢ ℤ = (Base‘ℤring) | |
6 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 5, 6 | rhmf 18549 | . . . . . 6 ⊢ ((ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅) → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅)) |
8 | ffn 5958 | . . . . . 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 2610 | . . . . . . 7 ⊢ (chr‘𝑅) = (chr‘𝑅) | |
11 | 10 | chrcl 19693 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (chr‘𝑅) ∈ ℕ0) |
12 | nn0z 11277 | . . . . . 6 ⊢ ((chr‘𝑅) ∈ ℕ0 → (chr‘𝑅) ∈ ℤ) | |
13 | 1, 11, 12 | 3syl 18 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑅) ∈ ℤ) |
14 | fvco2 6183 | . . . . 5 ⊢ (((ℤRHom‘𝑅) Fn ℤ ∧ (chr‘𝑅) ∈ ℤ) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅)))) | |
15 | 9, 13, 14 | syl2anc 691 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅)))) |
16 | eqid 2610 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | 10, 2, 16 | chrid 19694 | . . . . . 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 6107 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))) = (𝐹‘(0g‘𝑅))) |
20 | 15, 19 | eqtrd 2644 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘(0g‘𝑅))) |
21 | rhmco 18560 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅)) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤring RingHom 𝑆)) | |
22 | 4, 21 | mpdan 699 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤring RingHom 𝑆)) |
23 | rhmrcl2 18543 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring) | |
24 | eqid 2610 | . . . . . . 7 ⊢ (ℤRHom‘𝑆) = (ℤRHom‘𝑆) | |
25 | 24 | zrhrhmb 19678 | . . . . . 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 221 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆)) |
28 | 27 | fveq1d 6105 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = ((ℤRHom‘𝑆)‘(chr‘𝑅))) |
29 | rhmghm 18548 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
30 | eqid 2610 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
31 | 16, 30 | ghmid 17489 | . . . 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 2652 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0g‘𝑆)) |
34 | eqid 2610 | . . . 4 ⊢ (chr‘𝑆) = (chr‘𝑆) | |
35 | 34, 24, 30 | chrdvds 19695 | . . 3 ⊢ ((𝑆 ∈ Ring ∧ (chr‘𝑅) ∈ ℤ) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0g‘𝑆))) |
36 | 23, 13, 35 | syl2anc 691 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0g‘𝑆))) |
37 | 33, 36 | mpbird 246 | 1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℕ0cn0 11169 ℤcz 11254 ∥ cdvds 14821 Basecbs 15695 0gc0g 15923 GrpHom cghm 17480 Ringcrg 18370 RingHom crh 18535 ℤringzring 19637 ℤRHomczrh 19667 chrcchr 19669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-od 17771 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-rnghom 18538 df-subrg 18601 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-chr 19673 |
This theorem is referenced by: (None) |
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