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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 ℤ/nℤ 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 | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
cznrng.b | ⊢ 𝐵 = (Base‘𝑌) |
cznrng.x | ⊢ 𝑋 = (𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉) |
Ref | Expression |
---|---|
cznabel | ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵) → 𝑋 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 11708 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | 1 | adantr 473 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵) → 𝑁 ∈ ℕ0) |
3 | cznrng.y | . . . . 5 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
4 | 3 | zncrng 20383 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
5 | 2, 4 | syl 17 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵) → 𝑌 ∈ CRing) |
6 | crngring 19021 | . . 3 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
7 | ringabl 19043 | . . 3 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Abel) | |
8 | 5, 6, 7 | 3syl 18 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵) → 𝑌 ∈ Abel) |
9 | cznrng.x | . . . . 5 ⊢ 𝑋 = (𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉) | |
10 | 9 | fveq2i 6496 | . . . 4 ⊢ (Base‘𝑋) = (Base‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) |
11 | baseid 16389 | . . . . 5 ⊢ Base = Slot (Base‘ndx) | |
12 | basendxnmulrndx 16464 | . . . . 5 ⊢ (Base‘ndx) ≠ (.r‘ndx) | |
13 | 11, 12 | setsnid 16385 | . . . 4 ⊢ (Base‘𝑌) = (Base‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) |
14 | 10, 13 | eqtr4i 2799 | . . 3 ⊢ (Base‘𝑋) = (Base‘𝑌) |
15 | 9 | fveq2i 6496 | . . . 4 ⊢ (+g‘𝑋) = (+g‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) |
16 | plusgid 16442 | . . . . 5 ⊢ +g = Slot (+g‘ndx) | |
17 | plusgndxnmulrndx 16463 | . . . . 5 ⊢ (+g‘ndx) ≠ (.r‘ndx) | |
18 | 16, 17 | setsnid 16385 | . . . 4 ⊢ (+g‘𝑌) = (+g‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) |
19 | 15, 18 | eqtr4i 2799 | . . 3 ⊢ (+g‘𝑋) = (+g‘𝑌) |
20 | 14, 19 | ablprop 18667 | . 2 ⊢ (𝑋 ∈ Abel ↔ 𝑌 ∈ Abel) |
21 | 8, 20 | sylibr 226 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵) → 𝑋 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 〈cop 4441 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 ℕcn 11431 ℕ0cn0 11700 ndxcnx 16326 sSet csts 16327 Basecbs 16329 +gcplusg 16411 .rcmulr 16412 Abelcabl 18657 Ringcrg 19010 CRingccrg 19011 ℤ/nℤczn 20342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-addf 10406 ax-mulf 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-ec 8083 df-qs 8087 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-0g 16561 df-imas 16627 df-qus 16628 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-grp 17884 df-minusg 17885 df-sbg 17886 df-subg 18050 df-nsg 18051 df-eqg 18052 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-cring 19013 df-oppr 19086 df-subrg 19246 df-lmod 19348 df-lss 19416 df-lsp 19456 df-sra 19656 df-rgmod 19657 df-lidl 19658 df-rsp 19659 df-2idl 19716 df-cnfld 20238 df-zring 20310 df-zn 20346 |
This theorem is referenced by: cznrng 43530 |
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