Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrngnring | Structured version Visualization version GIF version |
Description: R is not a unital ring. (Contributed by AV, 6-Jan-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
2zrngbas.r | ⊢ 𝑅 = (ℂfld ↾s 𝐸) |
2zrngmmgm.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
2zrngnring | ⊢ 𝑅 ∉ Ring |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2zrng.e | . . . . . . 7 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
2 | 2zrngbas.r | . . . . . . 7 ⊢ 𝑅 = (ℂfld ↾s 𝐸) | |
3 | 2zrngmmgm.1 | . . . . . . 7 ⊢ 𝑀 = (mulGrp‘𝑅) | |
4 | 1, 2, 3 | 2zrngnmlid 41739 | . . . . . 6 ⊢ ∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 |
5 | 1, 2 | 2zrngbas 41726 | . . . . . . . . 9 ⊢ 𝐸 = (Base‘𝑅) |
6 | 3, 5 | mgpbas 18318 | . . . . . . . 8 ⊢ 𝐸 = (Base‘𝑀) |
7 | 1, 2 | 2zrngmul 41735 | . . . . . . . . 9 ⊢ · = (.r‘𝑅) |
8 | 3, 7 | mgpplusg 18316 | . . . . . . . 8 ⊢ · = (+g‘𝑀) |
9 | 6, 8 | isnmnd 17121 | . . . . . . 7 ⊢ (∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 → 𝑀 ∉ Mnd) |
10 | df-nel 2783 | . . . . . . 7 ⊢ (𝑀 ∉ Mnd ↔ ¬ 𝑀 ∈ Mnd) | |
11 | 9, 10 | sylib 207 | . . . . . 6 ⊢ (∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 → ¬ 𝑀 ∈ Mnd) |
12 | 4, 11 | ax-mp 5 | . . . . 5 ⊢ ¬ 𝑀 ∈ Mnd |
13 | 12 | 3mix2i 1227 | . . . 4 ⊢ (¬ 𝑅 ∈ Grp ∨ ¬ 𝑀 ∈ Mnd ∨ ¬ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥 · (𝑦(+g‘𝑅)𝑧)) = ((𝑥 · 𝑦)(+g‘𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝑅)(𝑦 · 𝑧)))) |
14 | 3ianor 1048 | . . . 4 ⊢ (¬ (𝑅 ∈ Grp ∧ 𝑀 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥 · (𝑦(+g‘𝑅)𝑧)) = ((𝑥 · 𝑦)(+g‘𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝑅)(𝑦 · 𝑧)))) ↔ (¬ 𝑅 ∈ Grp ∨ ¬ 𝑀 ∈ Mnd ∨ ¬ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥 · (𝑦(+g‘𝑅)𝑧)) = ((𝑥 · 𝑦)(+g‘𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝑅)(𝑦 · 𝑧))))) | |
15 | 13, 14 | mpbir 220 | . . 3 ⊢ ¬ (𝑅 ∈ Grp ∧ 𝑀 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥 · (𝑦(+g‘𝑅)𝑧)) = ((𝑥 · 𝑦)(+g‘𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝑅)(𝑦 · 𝑧)))) |
16 | eqid 2610 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
17 | eqid 2610 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
18 | 16, 3, 17, 7 | isring 18374 | . . 3 ⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝑀 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥 · (𝑦(+g‘𝑅)𝑧)) = ((𝑥 · 𝑦)(+g‘𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝑅)(𝑦 · 𝑧))))) |
19 | 15, 18 | mtbir 312 | . 2 ⊢ ¬ 𝑅 ∈ Ring |
20 | 19 | nelir 2886 | 1 ⊢ 𝑅 ∉ Ring |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∀wral 2896 ∃wrex 2897 {crab 2900 ‘cfv 5804 (class class class)co 6549 · cmul 9820 2c2 10947 ℤcz 11254 Basecbs 15695 ↾s cress 15696 +gcplusg 15768 Mndcmnd 17117 Grpcgrp 17245 mulGrpcmgp 18312 Ringcrg 18370 ℂfldccnfld 19567 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 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-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-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 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-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 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-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-mnd 17118 df-mgp 18313 df-ring 18372 df-cnfld 19568 |
This theorem is referenced by: (None) |
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