chrrhm - Metamath Proof Explorer

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Theorem chrrhm 21690

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 20564	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
2		eqid 2763	. . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
3	2	zrhrhm 21670	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅))
4	1, 3	syl 18	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅))
5		zringbas 21612	. . . . . . 7 ⊢ ℤ = (Base‘ℤ_ring)
6		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7	5, 6	rhmf 20572	. . . . . 6 ⊢ ((ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅) → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅))
8		ffn 6705	. . . . . 6 ⊢ ((ℤRHom‘𝑅):ℤ⟶(Base‘𝑅) → (ℤRHom‘𝑅) Fn ℤ)
9	4, 7, 8	3syl 19	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (ℤRHom‘𝑅) Fn ℤ)
10		eqid 2763	. . . . . . 7 ⊢ (chr‘𝑅) = (chr‘𝑅)
11	10	chrcl 21683	. . . . . 6 ⊢ (𝑅 ∈ Ring → (chr‘𝑅) ∈ ℕ₀)
12		nn0z 12619	. . . . . 6 ⊢ ((chr‘𝑅) ∈ ℕ₀ → (chr‘𝑅) ∈ ℤ)
13	1, 11, 12	3syl 19	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑅) ∈ ℤ)
14		fvco2 6978	. . . . 5 ⊢ (((ℤRHom‘𝑅) Fn ℤ ∧ (chr‘𝑅) ∈ ℤ) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))))
15	9, 13, 14	syl2anc 595	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))))
16		eqid 2763	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
17	10, 2, 16	chrid 21684	. . . . . 6 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘(chr‘𝑅)) = (0_g‘𝑅))
18	1, 17	syl 18	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((ℤRHom‘𝑅)‘(chr‘𝑅)) = (0_g‘𝑅))
19	18	fveq2d 6885	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))) = (𝐹‘(0_g‘𝑅)))
20	15, 19	eqtrd 2798	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘(0_g‘𝑅)))
21		rhmco 20596	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅)) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆))
22	4, 21	mpdan 699	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆))
23		rhmrcl2 20565	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
24		eqid 2763	. . . . . . 7 ⊢ (ℤRHom‘𝑆) = (ℤRHom‘𝑆)
25	24	zrhrhmb 21669	. . . . . 6 ⊢ (𝑆 ∈ Ring → ((𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆) ↔ (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆)))
26	23, 25	syl 18	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆) ↔ (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆)))
27	22, 26	mpbid 235	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆))
28	27	fveq1d 6883	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = ((ℤRHom‘𝑆)‘(chr‘𝑅)))
29		rhmghm 20571	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
30		eqid 2763	. . . . 5 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
31	16, 30	ghmid 19296	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
32	29, 31	syl 18	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
33	20, 28, 32	3eqtr3d 2806	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆))
34		eqid 2763	. . . 4 ⊢ (chr‘𝑆) = (chr‘𝑆)
35	34, 24, 30	chrdvds 21685	. . 3 ⊢ ((𝑆 ∈ Ring ∧ (chr‘𝑅) ∈ ℤ) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆)))
36	23, 13, 35	syl2anc 595	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆)))
37	33, 36	mpbird 260	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = wceq 1570 ∈ wcel 2143 class class class wbr 5109 ∘ ccom 5665 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℕ₀cn0 12508 ℤcz 12595 ∥ cdvds 16314 Basecbs 17273 0_gc0g 17496 GrpHom cghm 19287 Ringcrg 20319 RingHom crh 20556 ℤ_ringczring 21605 ℤRHomczrh 21658 chrcchr 21660

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183 ax-mulf 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-rp 13021 df-fz 13540 df-fl 13830 df-mod 13908 df-seq 14043 df-exp 14103 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-dvds 16315 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-0g 17498 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-mhm 18845 df-grp 19007 df-minusg 19008 df-sbg 19009 df-mulg 19138 df-subg 19193 df-ghm 19288 df-od 19602 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-cring 20322 df-rhm 20559 df-subrng 20654 df-subrg 20678 df-cnfld 21532 df-zring 21606 df-zrh 21662 df-chr 21664

This theorem is used by: (None)

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