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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 46241 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑋) |
6 | 1, 5 | mgpbas 19902 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑋)) |
7 | 4 | fveq2i 6845 | . . . . . . . 8 ⊢ (mulGrp‘𝑋) = (mulGrp‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) |
8 | 2 | fvexi 6856 | . . . . . . . . 9 ⊢ 𝑌 ∈ V |
9 | 3 | fvexi 6856 | . . . . . . . . . 10 ⊢ 𝐵 ∈ V |
10 | 9, 9 | mpoex 8012 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V |
11 | mulrid 17175 | . . . . . . . . . 10 ⊢ .r = Slot (.r‘ndx) | |
12 | 11 | setsid 17080 | . . . . . . . . 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 690 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) |
14 | 7, 13 | mgpplusg 19900 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (+g‘(mulGrp‘𝑋)) |
15 | 14 | eqcomi 2745 | . . . . . 6 ⊢ (+g‘(mulGrp‘𝑋)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) |
16 | simpr 485 | . . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ 𝐵) | |
17 | eluz2 12769 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) | |
18 | 1lt2 12324 | . . . . . . . . . 10 ⊢ 1 < 2 | |
19 | 1red 11156 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
20 | 2re 12227 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ | |
21 | 20 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
22 | zre 12503 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
23 | ltletr 11247 | . . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 < 𝑁)) | |
24 | 19, 21, 22, 23 | syl3anc 1371 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 < 𝑁)) |
25 | 24 | expcomd 417 | . . . . . . . . . . . 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 1111 | . . . . . . . . . 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 216 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁) |
30 | eluz2nn 12809 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
31 | 2, 3 | znhash 20965 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁) |
32 | 30, 31 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (♯‘𝐵) = 𝑁) |
33 | 29, 32 | breqtrrd 5133 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < (♯‘𝐵)) |
34 | 33 | adantr 481 | . . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 1 < (♯‘𝐵)) |
35 | 6, 15, 16, 34 | copisnmnd 46093 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → (mulGrp‘𝑋) ∉ Mnd) |
36 | df-nel 3050 | . . . . 5 ⊢ ((mulGrp‘𝑋) ∉ Mnd ↔ ¬ (mulGrp‘𝑋) ∈ Mnd) | |
37 | 35, 36 | sylib 217 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → ¬ (mulGrp‘𝑋) ∈ Mnd) |
38 | 37 | intn3an2d 1480 | . . 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 6845 | . . . 4 ⊢ (.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) = (.r‘𝑋) |
42 | 5, 1, 39, 41 | isring 19968 | . . 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 328 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → ¬ 𝑋 ∈ Ring) |
44 | df-nel 3050 | . 2 ⊢ (𝑋 ∉ Ring ↔ ¬ 𝑋 ∈ Ring) | |
45 | 43, 44 | sylibr 233 | 1 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝑋 ∉ Ring) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∉ wnel 3049 ∀wral 3064 Vcvv 3445 〈cop 4592 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 ℝcr 11050 1c1 11052 < clt 11189 ≤ cle 11190 ℕcn 12153 2c2 12208 ℤcz 12499 ℤ≥cuz 12763 ♯chash 14230 sSet csts 17035 ndxcnx 17065 Basecbs 17083 +gcplusg 17133 .rcmulr 17134 0gc0g 17321 Mndcmnd 18556 Grpcgrp 18748 mulGrpcmgp 19896 Ringcrg 19964 ℤ/nℤczn 20903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-hash 14231 df-dvds 16137 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-imas 17390 df-qus 17391 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 |
This theorem is referenced by: (None) |
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