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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 12428 | . . . . 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 20974 | . . . 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 19984 | . . 3 β’ (π β CRing β π β Ring) | |
7 | ringabl 20010 | . . 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 6849 | . . . 4 β’ (Baseβπ) = (Baseβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
11 | baseid 17094 | . . . . 5 β’ Base = Slot (Baseβndx) | |
12 | basendxnmulrndx 17184 | . . . . 5 β’ (Baseβndx) β (.rβndx) | |
13 | 11, 12 | setsnid 17089 | . . . 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 6849 | . . . 4 β’ (+gβπ) = (+gβ(π sSet β¨(.rβndx), (π₯ β π΅, π¦ β π΅ β¦ πΆ)β©)) |
16 | plusgid 17168 | . . . . 5 β’ +g = Slot (+gβndx) | |
17 | plusgndxnmulrndx 17186 | . . . . 5 β’ (+gβndx) β (.rβndx) | |
18 | 16, 17 | setsnid 17089 | . . . 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 19583 | . 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 4596 βcfv 6500 (class class class)co 7361 β cmpo 7363 βcn 12161 β0cn0 12421 sSet csts 17043 ndxcnx 17073 Basecbs 17091 +gcplusg 17141 .rcmulr 17142 Abelcabl 19571 Ringcrg 19972 CRingccrg 19973 β€/nβ€czn 20926 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-ec 8656 df-qs 8660 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-imas 17398 df-qus 17399 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-nsg 18934 df-eqg 18935 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-subrg 20262 df-lmod 20367 df-lss 20437 df-lsp 20477 df-sra 20678 df-rgmod 20679 df-lidl 20680 df-rsp 20681 df-2idl 20747 df-cnfld 20820 df-zring 20893 df-zn 20930 |
This theorem is referenced by: cznrng 46343 |
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