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Mirrors > Home > MPE Home > Th. List > chrcong | Structured version Visualization version GIF version |
Description: If two integers are congruent relative to the ring characteristic, their images in the ring are the same. (Contributed by Mario Carneiro, 24-Sep-2015.) |
Ref | Expression |
---|---|
chrcl.c | ⊢ 𝐶 = (chr‘𝑅) |
chrid.l | ⊢ 𝐿 = (ℤRHom‘𝑅) |
chrid.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
chrcong | ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ (𝑀 − 𝑁) ↔ (𝐿‘𝑀) = (𝐿‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ (od‘𝑅) = (od‘𝑅) | |
2 | eqid 2818 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | chrcl.c | . . . . 5 ⊢ 𝐶 = (chr‘𝑅) | |
4 | 1, 2, 3 | chrval 20600 | . . . 4 ⊢ ((od‘𝑅)‘(1r‘𝑅)) = 𝐶 |
5 | 4 | breq1i 5064 | . . 3 ⊢ (((od‘𝑅)‘(1r‘𝑅)) ∥ (𝑀 − 𝑁) ↔ 𝐶 ∥ (𝑀 − 𝑁)) |
6 | ringgrp 19231 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 6 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑅 ∈ Grp) |
8 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 2 | ringidcl 19247 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
10 | 9 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | simp2 1129 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ) | |
12 | simp3 1130 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
13 | eqid 2818 | . . . . 5 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
14 | chrid.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
15 | 8, 1, 13, 14 | odcong 18606 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (((od‘𝑅)‘(1r‘𝑅)) ∥ (𝑀 − 𝑁) ↔ (𝑀(.g‘𝑅)(1r‘𝑅)) = (𝑁(.g‘𝑅)(1r‘𝑅)))) |
16 | 7, 10, 11, 12, 15 | syl112anc 1366 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((od‘𝑅)‘(1r‘𝑅)) ∥ (𝑀 − 𝑁) ↔ (𝑀(.g‘𝑅)(1r‘𝑅)) = (𝑁(.g‘𝑅)(1r‘𝑅)))) |
17 | 5, 16 | syl5bbr 286 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ (𝑀 − 𝑁) ↔ (𝑀(.g‘𝑅)(1r‘𝑅)) = (𝑁(.g‘𝑅)(1r‘𝑅)))) |
18 | chrid.l | . . . . 5 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
19 | 18, 13, 2 | zrhmulg 20585 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ) → (𝐿‘𝑀) = (𝑀(.g‘𝑅)(1r‘𝑅))) |
20 | 19 | 3adant3 1124 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐿‘𝑀) = (𝑀(.g‘𝑅)(1r‘𝑅))) |
21 | 18, 13, 2 | zrhmulg 20585 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐿‘𝑁) = (𝑁(.g‘𝑅)(1r‘𝑅))) |
22 | 21 | 3adant2 1123 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐿‘𝑁) = (𝑁(.g‘𝑅)(1r‘𝑅))) |
23 | 20, 22 | eqeq12d 2834 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐿‘𝑀) = (𝐿‘𝑁) ↔ (𝑀(.g‘𝑅)(1r‘𝑅)) = (𝑁(.g‘𝑅)(1r‘𝑅)))) |
24 | 17, 23 | bitr4d 283 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ (𝑀 − 𝑁) ↔ (𝐿‘𝑀) = (𝐿‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 − cmin 10858 ℤcz 11969 ∥ cdvds 15595 Basecbs 16471 0gc0g 16701 Grpcgrp 18041 .gcmg 18162 odcod 18581 1rcur 19180 Ringcrg 19226 ℤRHomczrh 20575 chrcchr 20577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-od 18585 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-rnghom 19396 df-subrg 19462 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-chr 20581 |
This theorem is referenced by: (None) |
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