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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 12477 | . . . . 5 β’ (π β β β π β β0) | |
2 | 1 | adantr 480 | . . . 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 21409 | . . . 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 20142 | . . 3 β’ (π β CRing β π β Ring) | |
7 | ringabl 20172 | . . 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 6885 | . . . 4 β’ (Baseβπ) = (Baseβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
11 | baseid 17148 | . . . . 5 β’ Base = Slot (Baseβndx) | |
12 | basendxnmulrndx 17241 | . . . . 5 β’ (Baseβndx) β (.rβndx) | |
13 | 11, 12 | setsnid 17143 | . . . 4 β’ (Baseβπ) = (Baseβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
14 | 10, 13 | eqtr4i 2755 | . . 3 β’ (Baseβπ) = (Baseβπ) |
15 | 9 | fveq2i 6885 | . . . 4 β’ (+gβπ) = (+gβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
16 | plusgid 17225 | . . . . 5 β’ +g = Slot (+gβndx) | |
17 | plusgndxnmulrndx 17243 | . . . . 5 β’ (+gβndx) β (.rβndx) | |
18 | 16, 17 | setsnid 17143 | . . . 4 β’ (+gβπ) = (+gβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
19 | 15, 18 | eqtr4i 2755 | . . 3 β’ (+gβπ) = (+gβπ) |
20 | 14, 19 | ablprop 19705 | . 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 395 = wceq 1533 β wcel 2098 β¨cop 4627 βcfv 6534 (class class class)co 7402 β cmpo 7404 βcn 12210 β0cn0 12470 sSet csts 17097 ndxcnx 17127 Basecbs 17145 +gcplusg 17198 .rcmulr 17199 Abelcabl 19693 Ringcrg 20130 CRingccrg 20131 β€/nβ€czn 21359 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-ec 8702 df-qs 8706 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-starv 17213 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-0g 17388 df-imas 17455 df-qus 17456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-nsg 19043 df-eqg 19044 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-cring 20133 df-oppr 20228 df-subrng 20438 df-subrg 20463 df-lmod 20700 df-lss 20771 df-lsp 20811 df-sra 21013 df-rgmod 21014 df-lidl 21059 df-rsp 21060 df-2idl 21099 df-cnfld 21231 df-zring 21304 df-zn 21363 |
This theorem is referenced by: cznrng 47149 |
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