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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 2610 | . . . . . . 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 41745 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑋) |
6 | 1, 5 | mgpbas 18318 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑋)) |
7 | 4 | fveq2i 6106 | . . . . . . . 8 ⊢ (mulGrp‘𝑋) = (mulGrp‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) |
8 | fvex 6113 | . . . . . . . . . 10 ⊢ (ℤ/nℤ‘𝑁) ∈ V | |
9 | 2, 8 | eqeltri 2684 | . . . . . . . . 9 ⊢ 𝑌 ∈ V |
10 | fvex 6113 | . . . . . . . . . . 11 ⊢ (Base‘𝑌) ∈ V | |
11 | 3, 10 | eqeltri 2684 | . . . . . . . . . 10 ⊢ 𝐵 ∈ V |
12 | 11, 11 | mpt2ex 7136 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V |
13 | mulrid 15822 | . . . . . . . . . 10 ⊢ .r = Slot (.r‘ndx) | |
14 | 13 | setsid 15742 | . . . . . . . . 9 ⊢ ((𝑌 ∈ V ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉))) |
15 | 9, 12, 14 | mp2an 704 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) |
16 | 7, 15 | mgpplusg 18316 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (+g‘(mulGrp‘𝑋)) |
17 | 16 | eqcomi 2619 | . . . . . 6 ⊢ (+g‘(mulGrp‘𝑋)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) |
18 | simpr 476 | . . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ 𝐵) | |
19 | eluz2 11569 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) | |
20 | 1lt2 11071 | . . . . . . . . . 10 ⊢ 1 < 2 | |
21 | 1red 9934 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
22 | 2re 10967 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ | |
23 | 22 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
24 | zre 11258 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
25 | ltletr 10008 | . . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 < 𝑁)) | |
26 | 21, 23, 24, 25 | syl3anc 1318 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 < 𝑁)) |
27 | 26 | expcomd 453 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (2 ≤ 𝑁 → (1 < 2 → 1 < 𝑁))) |
28 | 27 | a1i 11 | . . . . . . . . . . 11 ⊢ (2 ∈ ℤ → (𝑁 ∈ ℤ → (2 ≤ 𝑁 → (1 < 2 → 1 < 𝑁)))) |
29 | 28 | 3imp 1249 | . . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁) → (1 < 2 → 1 < 𝑁)) |
30 | 20, 29 | mpi 20 | . . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁) → 1 < 𝑁) |
31 | 19, 30 | sylbi 206 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁) |
32 | eluz2nn 11602 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
33 | 2, 3 | znhash 19726 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (#‘𝐵) = 𝑁) |
34 | 32, 33 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (#‘𝐵) = 𝑁) |
35 | 31, 34 | breqtrrd 4611 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < (#‘𝐵)) |
36 | 35 | adantr 480 | . . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 1 < (#‘𝐵)) |
37 | 6, 17, 18, 36 | copisnmnd 41599 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → (mulGrp‘𝑋) ∉ Mnd) |
38 | df-nel 2783 | . . . . 5 ⊢ ((mulGrp‘𝑋) ∉ Mnd ↔ ¬ (mulGrp‘𝑋) ∈ Mnd) | |
39 | 37, 38 | sylib 207 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → ¬ (mulGrp‘𝑋) ∈ Mnd) |
40 | 39 | intn3an2d 1435 | . . 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))))) |
41 | eqid 2610 | . . . 4 ⊢ (+g‘𝑋) = (+g‘𝑋) | |
42 | 4 | eqcomi 2619 | . . . . 5 ⊢ (𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉) = 𝑋 |
43 | 42 | fveq2i 6106 | . . . 4 ⊢ (.r‘(𝑌 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)〉)) = (.r‘𝑋) |
44 | 5, 1, 41, 43 | isring 18374 | . . 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))))) |
45 | 40, 44 | sylnibr 318 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → ¬ 𝑋 ∈ Ring) |
46 | df-nel 2783 | . 2 ⊢ (𝑋 ∉ Ring ↔ ¬ 𝑋 ∈ Ring) | |
47 | 45, 46 | sylibr 223 | 1 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝑋 ∉ Ring) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 ∀wral 2896 Vcvv 3173 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℝcr 9814 1c1 9816 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 ℤcz 11254 ℤ≥cuz 11563 #chash 12979 ndxcnx 15692 sSet csts 15693 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 0gc0g 15923 Mndcmnd 17117 Grpcgrp 17245 mulGrpcmgp 18312 Ringcrg 18370 ℤ/nℤczn 19670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-hash 12980 df-dvds 14822 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 |
This theorem is referenced by: (None) |
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