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Theorem chrrhm 21473

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 20396	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
2		eqid 2729	. . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
3	2	zrhrhm 21453	. . . . . . 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 21395	. . . . . . 7 ⊢ ℤ = (Base‘ℤ_ring)
6		eqid 2729	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7	5, 6	rhmf 20405	. . . . . 6 ⊢ ((ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅) → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅))
8		ffn 6670	. . . . . 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 2729	. . . . . . 7 ⊢ (chr‘𝑅) = (chr‘𝑅)
11	10	chrcl 21466	. . . . . 6 ⊢ (𝑅 ∈ Ring → (chr‘𝑅) ∈ ℕ₀)
12		nn0z 12530	. . . . . 6 ⊢ ((chr‘𝑅) ∈ ℕ₀ → (chr‘𝑅) ∈ ℤ)
13	1, 11, 12	3syl 18	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑅) ∈ ℤ)
14		fvco2 6940	. . . . 5 ⊢ (((ℤRHom‘𝑅) Fn ℤ ∧ (chr‘𝑅) ∈ ℤ) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))))
15	9, 13, 14	syl2anc 584	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))))
16		eqid 2729	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
17	10, 2, 16	chrid 21467	. . . . . 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 6844	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘((ℤRHom‘𝑅)‘(chr‘𝑅))) = (𝐹‘(0_g‘𝑅)))
20	15, 19	eqtrd 2764	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = (𝐹‘(0_g‘𝑅)))
21		rhmco 20421	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (ℤRHom‘𝑅) ∈ (ℤ_ring RingHom 𝑅)) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆))
22	4, 21	mpdan 687	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) ∈ (ℤ_ring RingHom 𝑆))
23		rhmrcl2 20397	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
24		eqid 2729	. . . . . . 7 ⊢ (ℤRHom‘𝑆) = (ℤRHom‘𝑆)
25	24	zrhrhmb 21452	. . . . . 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 232	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∘ (ℤRHom‘𝑅)) = (ℤRHom‘𝑆))
28	27	fveq1d 6842	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 ∘ (ℤRHom‘𝑅))‘(chr‘𝑅)) = ((ℤRHom‘𝑆)‘(chr‘𝑅)))
29		rhmghm 20404	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
30		eqid 2729	. . . . 5 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
31	16, 30	ghmid 19136	. . . 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 2772	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆))
34		eqid 2729	. . . 4 ⊢ (chr‘𝑆) = (chr‘𝑆)
35	34, 24, 30	chrdvds 21468	. . 3 ⊢ ((𝑆 ∈ Ring ∧ (chr‘𝑅) ∈ ℤ) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆)))
36	23, 13, 35	syl2anc 584	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((chr‘𝑆) ∥ (chr‘𝑅) ↔ ((ℤRHom‘𝑆)‘(chr‘𝑅)) = (0_g‘𝑆)))
37	33, 36	mpbird 257	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2109 class class class wbr 5102 ∘ ccom 5635 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℕ₀cn0 12418 ℤcz 12505 ∥ cdvds 16198 Basecbs 17155 0_gc0g 17378 GrpHom cghm 19126 Ringcrg 20153 RingHom crh 20389 ℤ_ringczring 21388 ℤRHomczrh 21441 chrcchr 21443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-rp 12928 df-fz 13445 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16199 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17380 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-ghm 19127 df-od 19442 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-rhm 20392 df-subrng 20466 df-subrg 20490 df-cnfld 21297 df-zring 21389 df-zrh 21445 df-chr 21447

This theorem is referenced by: (None)

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