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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 47892, and even a non-unital ring, see 2zrng 47883. (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 47813 | . 2 ⊢ 1 ∈ 𝑂 |
3 | 1 | 2nodd 47814 | . . 3 ⊢ 2 ∉ 𝑂 |
4 | 1p1e2 12414 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | neleq1 3054 | . . . 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 231 | . 2 ⊢ (1 + 1) ∉ 𝑂 |
8 | oddinmgm.r | . . . 4 ⊢ 𝑀 = (ℂfld ↾s 𝑂) | |
9 | 1, 8 | oddibas 47815 | . . 3 ⊢ 𝑂 = (Base‘𝑀) |
10 | 1, 8 | oddiadd 47816 | . . 3 ⊢ + = (+g‘𝑀) |
11 | 9, 10 | isnmgm 18677 | . 2 ⊢ ((1 ∈ 𝑂 ∧ 1 ∈ 𝑂 ∧ (1 + 1) ∉ 𝑂) → 𝑀 ∉ Mgm) |
12 | 2, 2, 7, 11 | mp3an 1461 | 1 ⊢ 𝑀 ∉ Mgm |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2103 ∉ wnel 3048 ∃wrex 3072 {crab 3438 (class class class)co 7445 1c1 11181 + caddc 11183 · cmul 11185 2c2 12344 ℤcz 12635 ↾s cress 17282 Mgmcmgm 18671 ℂfldccnfld 21382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-addf 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-mgm 18673 df-cnfld 21383 |
This theorem is referenced by: (None) |
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