chrrhm - Metamath Proof Explorer

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Theorem chrrhm 20647

Description: The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)

**Assertion**
Ref	Expression
chrrhm	⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))

**Proof of Theorem chrrhm**
Step	Hyp	Ref	Expression
1		rhmrcl1 19878	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
2		eqid 2738	. . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
3	2	zrhrhm 20625	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅))
4	1, 3	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅))
5		zringbas 20588	. . . . . . 7 ⊢ ℤ = (Base‘ℤ_ring)
6		eqid 2738	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7	5, 6	rhmf 19885	. . . . . 6 ⊢ ((ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅) → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅))
8		ffn 6584	. . . . . 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 2738	. . . . . . 7 ⊢ (chr‘𝑅) = (chr‘𝑅)
11	10	chrcl 20642	. . . . . 6 ⊢ (𝑅 ∈ Ring → (chr‘𝑅) ∈ ℕ₀)
12		nn0z 12273	. . . . . 6 ⊢ ((chr‘𝑅) ∈ ℕ₀ → (chr‘𝑅) ∈ ℤ)
13	1, 11, 12	3syl 18	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑅) ∈ ℤ)
14		fvco2 6847	. . . . 5 ⊢ (((ℤRHom‘𝑅) Fn ℤ ∧ (chr‘𝑅) ∈ ℤ) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))))
15	9, 13, 14	syl2anc 583	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))))
16		eqid 2738	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
17	10, 2, 16	chrid 20643	. . . . . 6 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘(chr‘𝑅)) = (0_g‘𝑅))
18	1, 17	syl 17	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((ℤRHom‘𝑅)‘(chr‘𝑅)) = (0_g‘𝑅))
19	18	fveq2d 6760	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))) = (𝐹‘(0_g‘𝑅)))
20	15, 19	eqtrd 2778	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘(0_g‘𝑅)))
21		rhmco 19896	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅)) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆))
22	4, 21	mpdan 683	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆))
23		rhmrcl2 19879	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
24		eqid 2738	. . . . . . 7 ⊢ (ℤRHom‘𝑆) = (ℤRHom‘𝑆)
25	24	zrhrhmb 20624	. . . . . 6 ⊢ (𝑆 ∈ Ring → ((𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆) ↔ (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆)))
26	23, 25	syl 17	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆) ↔ (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆)))
27	22, 26	mpbid 231	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆))
28	27	fveq1d 6758	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = ((ℤRHom‘𝑆)‘(chr‘𝑅)))
29		rhmghm 19884	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
30		eqid 2738	. . . . 5 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
31	16, 30	ghmid 18755	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
32	29, 31	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
33	20, 28, 32	3eqtr3d 2786	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆))
34		eqid 2738	. . . 4 ⊢ (chr‘𝑆) = (chr‘𝑆)
35	34, 24, 30	chrdvds 20644	. . 3 ⊢ ((𝑆 ∈ Ring ∧ (chr‘𝑅) ∈ ℤ) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆)))
36	23, 13, 35	syl2anc 583	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆)))
37	33, 36	mpbird 256	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ∘ ccom 5584 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℕ₀cn0 12163 ℤcz 12249 ∥ cdvds 15891 Basecbs 16840 0_gc0g 17067 GrpHom cghm 18746 Ringcrg 19698 RingHom crh 19871 ℤ_ringzring 20582 ℤRHomczrh 20613 chrcchr 20615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-od 19051 df-cmn 19303 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-rnghom 19874 df-subrg 19937 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-chr 20619

This theorem is referenced by: (None)

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