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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 | β’ πΆ = (chrβπ ) |
chrid.l | β’ πΏ = (β€RHomβπ ) |
chrid.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
chrcong | β’ ((π β Ring β§ π β β€ β§ π β β€) β (πΆ β₯ (π β π) β (πΏβπ) = (πΏβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . 5 β’ (odβπ ) = (odβπ ) | |
2 | eqid 2726 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
3 | chrcl.c | . . . . 5 β’ πΆ = (chrβπ ) | |
4 | 1, 2, 3 | chrval 21409 | . . . 4 β’ ((odβπ )β(1rβπ )) = πΆ |
5 | 4 | breq1i 5148 | . . 3 β’ (((odβπ )β(1rβπ )) β₯ (π β π) β πΆ β₯ (π β π)) |
6 | ringgrp 20140 | . . . . 5 β’ (π β Ring β π β Grp) | |
7 | 6 | 3ad2ant1 1130 | . . . 4 β’ ((π β Ring β§ π β β€ β§ π β β€) β π β Grp) |
8 | eqid 2726 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
9 | 8, 2 | ringidcl 20162 | . . . . 5 β’ (π β Ring β (1rβπ ) β (Baseβπ )) |
10 | 9 | 3ad2ant1 1130 | . . . 4 β’ ((π β Ring β§ π β β€ β§ π β β€) β (1rβπ ) β (Baseβπ )) |
11 | simp2 1134 | . . . 4 β’ ((π β Ring β§ π β β€ β§ π β β€) β π β β€) | |
12 | simp3 1135 | . . . 4 β’ ((π β Ring β§ π β β€ β§ π β β€) β π β β€) | |
13 | eqid 2726 | . . . . 5 β’ (.gβπ ) = (.gβπ ) | |
14 | chrid.z | . . . . 5 β’ 0 = (0gβπ ) | |
15 | 8, 1, 13, 14 | odcong 19466 | . . . 4 β’ ((π β Grp β§ (1rβπ ) β (Baseβπ ) β§ (π β β€ β§ π β β€)) β (((odβπ )β(1rβπ )) β₯ (π β π) β (π(.gβπ )(1rβπ )) = (π(.gβπ )(1rβπ )))) |
16 | 7, 10, 11, 12, 15 | syl112anc 1371 | . . 3 β’ ((π β Ring β§ π β β€ β§ π β β€) β (((odβπ )β(1rβπ )) β₯ (π β π) β (π(.gβπ )(1rβπ )) = (π(.gβπ )(1rβπ )))) |
17 | 5, 16 | bitr3id 285 | . 2 β’ ((π β Ring β§ π β β€ β§ π β β€) β (πΆ β₯ (π β π) β (π(.gβπ )(1rβπ )) = (π(.gβπ )(1rβπ )))) |
18 | chrid.l | . . . . 5 β’ πΏ = (β€RHomβπ ) | |
19 | 18, 13, 2 | zrhmulg 21391 | . . . 4 β’ ((π β Ring β§ π β β€) β (πΏβπ) = (π(.gβπ )(1rβπ ))) |
20 | 19 | 3adant3 1129 | . . 3 β’ ((π β Ring β§ π β β€ β§ π β β€) β (πΏβπ) = (π(.gβπ )(1rβπ ))) |
21 | 18, 13, 2 | zrhmulg 21391 | . . . 4 β’ ((π β Ring β§ π β β€) β (πΏβπ) = (π(.gβπ )(1rβπ ))) |
22 | 21 | 3adant2 1128 | . . 3 β’ ((π β Ring β§ π β β€ β§ π β β€) β (πΏβπ) = (π(.gβπ )(1rβπ ))) |
23 | 20, 22 | eqeq12d 2742 | . 2 β’ ((π β Ring β§ π β β€ β§ π β β€) β ((πΏβπ) = (πΏβπ) β (π(.gβπ )(1rβπ )) = (π(.gβπ )(1rβπ )))) |
24 | 17, 23 | bitr4d 282 | 1 β’ ((π β Ring β§ π β β€ β§ π β β€) β (πΆ β₯ (π β π) β (πΏβπ) = (πΏβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 (class class class)co 7404 β cmin 11445 β€cz 12559 β₯ cdvds 16201 Basecbs 17150 0gc0g 17391 Grpcgrp 18860 .gcmg 18992 odcod 19441 1rcur 20083 Ringcrg 20135 β€RHomczrh 21381 chrcchr 21383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-dvds 16202 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-ghm 19136 df-od 19445 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-rhm 20371 df-subrng 20443 df-subrg 20468 df-cnfld 21236 df-zring 21329 df-zrh 21385 df-chr 21387 |
This theorem is referenced by: (None) |
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