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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cznabel | Structured version Visualization version GIF version |
Description: The ring constructed from a β€/nβ€ structure by replacing the (multiplicative) ring operation by a constant operation is an abelian group. (Contributed by AV, 16-Feb-2020.) |
Ref | Expression |
---|---|
cznrng.y | β’ π = (β€/nβ€βπ) |
cznrng.b | β’ π΅ = (Baseβπ) |
cznrng.x | β’ π = (π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©) |
Ref | Expression |
---|---|
cznabel | β’ ((π β β β§ πΆ β π΅) β π β Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 12479 | . . . . 5 β’ (π β β β π β β0) | |
2 | 1 | adantr 482 | . . . 4 β’ ((π β β β§ πΆ β π΅) β π β β0) |
3 | cznrng.y | . . . . 5 β’ π = (β€/nβ€βπ) | |
4 | 3 | zncrng 21100 | . . . 4 β’ (π β β0 β π β CRing) |
5 | 2, 4 | syl 17 | . . 3 β’ ((π β β β§ πΆ β π΅) β π β CRing) |
6 | crngring 20068 | . . 3 β’ (π β CRing β π β Ring) | |
7 | ringabl 20098 | . . 3 β’ (π β Ring β π β Abel) | |
8 | 5, 6, 7 | 3syl 18 | . 2 β’ ((π β β β§ πΆ β π΅) β π β Abel) |
9 | cznrng.x | . . . . 5 β’ π = (π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©) | |
10 | 9 | fveq2i 6895 | . . . 4 β’ (Baseβπ) = (Baseβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
11 | baseid 17147 | . . . . 5 β’ Base = Slot (Baseβndx) | |
12 | basendxnmulrndx 17240 | . . . . 5 β’ (Baseβndx) β (.rβndx) | |
13 | 11, 12 | setsnid 17142 | . . . 4 β’ (Baseβπ) = (Baseβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
14 | 10, 13 | eqtr4i 2764 | . . 3 β’ (Baseβπ) = (Baseβπ) |
15 | 9 | fveq2i 6895 | . . . 4 β’ (+gβπ) = (+gβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
16 | plusgid 17224 | . . . . 5 β’ +g = Slot (+gβndx) | |
17 | plusgndxnmulrndx 17242 | . . . . 5 β’ (+gβndx) β (.rβndx) | |
18 | 16, 17 | setsnid 17142 | . . . 4 β’ (+gβπ) = (+gβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
19 | 15, 18 | eqtr4i 2764 | . . 3 β’ (+gβπ) = (+gβπ) |
20 | 14, 19 | ablprop 19661 | . 2 β’ (π β Abel β π β Abel) |
21 | 8, 20 | sylibr 233 | 1 β’ ((π β β β§ πΆ β π΅) β π β Abel) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β¨cop 4635 βcfv 6544 (class class class)co 7409 β cmpo 7411 βcn 12212 β0cn0 12472 sSet csts 17096 ndxcnx 17126 Basecbs 17144 +gcplusg 17197 .rcmulr 17198 Abelcabl 19649 Ringcrg 20056 CRingccrg 20057 β€/nβ€czn 21052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-ec 8705 df-qs 8709 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-0g 17387 df-imas 17454 df-qus 17455 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-nsg 19004 df-eqg 19005 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-subrg 20317 df-lmod 20473 df-lss 20543 df-lsp 20583 df-sra 20785 df-rgmod 20786 df-lidl 20787 df-rsp 20788 df-2idl 20857 df-cnfld 20945 df-zring 21018 df-zn 21056 |
This theorem is referenced by: cznrng 46853 |
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