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Mirrors > Home > MPE Home > Th. List > Mathboxes > oddinmgm | Structured version Visualization version GIF version |
Description: The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 42469, and even a non-unital ring, see 2zrng 42460. (Contributed by AV, 3-Feb-2020.) |
Ref | Expression |
---|---|
oddinmgm.e | ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} |
oddinmgm.r | ⊢ 𝑀 = (ℂfld ↾s 𝑂) |
Ref | Expression |
---|---|
oddinmgm | ⊢ 𝑀 ∉ Mgm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddinmgm.e | . . 3 ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} | |
2 | 1 | 1odd 42336 | . 2 ⊢ 1 ∈ 𝑂 |
3 | 1 | 2nodd 42337 | . . 3 ⊢ 2 ∉ 𝑂 |
4 | 1p1e2 11337 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | neleq1 3051 | . . . 4 ⊢ ((1 + 1) = 2 → ((1 + 1) ∉ 𝑂 ↔ 2 ∉ 𝑂)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((1 + 1) ∉ 𝑂 ↔ 2 ∉ 𝑂) |
7 | 3, 6 | mpbir 221 | . 2 ⊢ (1 + 1) ∉ 𝑂 |
8 | oddinmgm.r | . . . 4 ⊢ 𝑀 = (ℂfld ↾s 𝑂) | |
9 | 1, 8 | oddibas 42338 | . . 3 ⊢ 𝑂 = (Base‘𝑀) |
10 | 1, 8 | oddiadd 42339 | . . 3 ⊢ + = (+g‘𝑀) |
11 | 9, 10 | isnmgm 17450 | . 2 ⊢ ((1 ∈ 𝑂 ∧ 1 ∈ 𝑂 ∧ (1 + 1) ∉ 𝑂) → 𝑀 ∉ Mgm) |
12 | 2, 2, 7, 11 | mp3an 1572 | 1 ⊢ 𝑀 ∉ Mgm |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 ∉ wnel 3046 ∃wrex 3062 {crab 3065 (class class class)co 6792 1c1 10139 + caddc 10141 · cmul 10143 2c2 11272 ℤcz 11580 ↾s cress 16061 Mgmcmgm 17444 ℂfldccnfld 19957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-addf 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11496 df-z 11581 df-dec 11697 df-uz 11890 df-fz 12530 df-struct 16062 df-ndx 16063 df-slot 16064 df-base 16066 df-sets 16067 df-ress 16068 df-plusg 16158 df-mulr 16159 df-starv 16160 df-tset 16164 df-ple 16165 df-ds 16168 df-unif 16169 df-mgm 17446 df-cnfld 19958 |
This theorem is referenced by: (None) |
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