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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cznnring | Structured version Visualization version GIF version | ||
| Description: The ring constructed from a ℤ/nℤ structure with 1 < 𝑛 by replacing the (multiplicative) ring operation by a constant operation is not a unital ring. (Contributed by AV, 17-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)〉) | 
| cznrng.0 | ⊢ 0 = (0g‘𝑌) | 
| Ref | Expression | 
|---|---|
| cznnring | ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝑋 ∉ Ring) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . . . . . 7 ⊢ (mulGrp‘𝑋) = (mulGrp‘𝑋) | |
| 2 | cznrng.y | . . . . . . . 8 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
| 3 | cznrng.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌) | |
| 4 | cznrng.x | . . . . . . . 8 ⊢ 𝑋 = (𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉) | |
| 5 | 2, 3, 4 | cznrnglem 48180 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑋) | 
| 6 | 1, 5 | mgpbas 20143 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑋)) | 
| 7 | 4 | fveq2i 6908 | . . . . . . . 8 ⊢ (mulGrp‘𝑋) = (mulGrp‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) | 
| 8 | 2 | fvexi 6919 | . . . . . . . . 9 ⊢ 𝑌 ∈ V | 
| 9 | 3 | fvexi 6919 | . . . . . . . . . 10 ⊢ 𝐵 ∈ V | 
| 10 | 9, 9 | mpoex 8105 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | 
| 11 | mulridx 17339 | . . . . . . . . . 10 ⊢ .r = Slot (.r‘ndx) | |
| 12 | 11 | setsid 17245 | . . . . . . . . 9 ⊢ ((𝑌 ∈ V ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))) | 
| 13 | 8, 10, 12 | mp2an 692 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) | 
| 14 | 7, 13 | mgpplusg 20142 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (+g‘(mulGrp‘𝑋)) | 
| 15 | 14 | eqcomi 2745 | . . . . . 6 ⊢ (+g‘(mulGrp‘𝑋)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) | 
| 16 | simpr 484 | . . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ 𝐵) | |
| 17 | eluz2 12885 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) | |
| 18 | 1lt2 12438 | . . . . . . . . . 10 ⊢ 1 < 2 | |
| 19 | 1red 11263 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
| 20 | 2re 12341 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ | |
| 21 | 20 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) | 
| 22 | zre 12619 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
| 23 | ltletr 11354 | . . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 < 𝑁)) | |
| 24 | 19, 21, 22, 23 | syl3anc 1372 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 < 𝑁)) | 
| 25 | 24 | expcomd 416 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (2 ≤ 𝑁 → (1 < 2 → 1 < 𝑁))) | 
| 26 | 25 | a1i 11 | . . . . . . . . . . 11 ⊢ (2 ∈ ℤ → (𝑁 ∈ ℤ → (2 ≤ 𝑁 → (1 < 2 → 1 < 𝑁)))) | 
| 27 | 26 | 3imp 1110 | . . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁) → (1 < 2 → 1 < 𝑁)) | 
| 28 | 18, 27 | mpi 20 | . . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁) → 1 < 𝑁) | 
| 29 | 17, 28 | sylbi 217 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁) | 
| 30 | eluz2nn 12925 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
| 31 | 2, 3 | znhash 21578 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁) | 
| 32 | 30, 31 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (♯‘𝐵) = 𝑁) | 
| 33 | 29, 32 | breqtrrd 5170 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < (♯‘𝐵)) | 
| 34 | 33 | adantr 480 | . . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 1 < (♯‘𝐵)) | 
| 35 | 6, 15, 16, 34 | copisnmnd 48090 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → (mulGrp‘𝑋) ∉ Mnd) | 
| 36 | df-nel 3046 | . . . . 5 ⊢ ((mulGrp‘𝑋) ∉ Mnd ↔ ¬ (mulGrp‘𝑋) ∈ Mnd) | |
| 37 | 35, 36 | sylib 218 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → ¬ (mulGrp‘𝑋) ∈ Mnd) | 
| 38 | 37 | intn3an2d 1481 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → ¬ (𝑋 ∈ Grp ∧ (mulGrp‘𝑋) ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))(𝑏(+g‘𝑋)𝑐)) = ((𝑎(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑏)(+g‘𝑋)(𝑎(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑐)) ∧ ((𝑎(+g‘𝑋)𝑏)(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑐) = ((𝑎(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑐)(+g‘𝑋)(𝑏(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑐))))) | 
| 39 | eqid 2736 | . . . 4 ⊢ (+g‘𝑋) = (+g‘𝑋) | |
| 40 | 4 | eqcomi 2745 | . . . . 5 ⊢ (𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉) = 𝑋 | 
| 41 | 40 | fveq2i 6908 | . . . 4 ⊢ (.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) = (.r‘𝑋) | 
| 42 | 5, 1, 39, 41 | isring 20235 | . . 3 ⊢ (𝑋 ∈ Ring ↔ (𝑋 ∈ Grp ∧ (mulGrp‘𝑋) ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))(𝑏(+g‘𝑋)𝑐)) = ((𝑎(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑏)(+g‘𝑋)(𝑎(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑐)) ∧ ((𝑎(+g‘𝑋)𝑏)(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑐) = ((𝑎(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑐)(+g‘𝑋)(𝑏(.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))𝑐))))) | 
| 43 | 38, 42 | sylnibr 329 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → ¬ 𝑋 ∈ Ring) | 
| 44 | df-nel 3046 | . 2 ⊢ (𝑋 ∉ Ring ↔ ¬ 𝑋 ∈ Ring) | |
| 45 | 43, 44 | sylibr 234 | 1 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝑋 ∉ Ring) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∉ wnel 3045 ∀wral 3060 Vcvv 3479 〈cop 4631 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ℝcr 11155 1c1 11157 < clt 11296 ≤ cle 11297 ℕcn 12267 2c2 12322 ℤcz 12615 ℤ≥cuz 12879 ♯chash 14370 sSet csts 17201 ndxcnx 17231 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 0gc0g 17485 Mndcmnd 18748 Grpcgrp 18952 mulGrpcmgp 20138 Ringcrg 20231 ℤ/nℤczn 21514 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-hash 14371 df-dvds 16292 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17487 df-imas 17554 df-qus 17555 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-lmod 20861 df-lss 20931 df-lsp 20971 df-sra 21173 df-rgmod 21174 df-lidl 21219 df-rsp 21220 df-2idl 21261 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-zn 21518 | 
| This theorem is referenced by: (None) | 
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