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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 47113, and even a non-unital ring, see 2zrng 47104. (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 47034 | . 2 ⊢ 1 ∈ 𝑂 |
3 | 1 | 2nodd 47035 | . . 3 ⊢ 2 ∉ 𝑂 |
4 | 1p1e2 12334 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | neleq1 3044 | . . . 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 230 | . 2 ⊢ (1 + 1) ∉ 𝑂 |
8 | oddinmgm.r | . . . 4 ⊢ 𝑀 = (ℂfld ↾s 𝑂) | |
9 | 1, 8 | oddibas 47036 | . . 3 ⊢ 𝑂 = (Base‘𝑀) |
10 | 1, 8 | oddiadd 47037 | . . 3 ⊢ + = (+g‘𝑀) |
11 | 9, 10 | isnmgm 18567 | . 2 ⊢ ((1 ∈ 𝑂 ∧ 1 ∈ 𝑂 ∧ (1 + 1) ∉ 𝑂) → 𝑀 ∉ Mgm) |
12 | 2, 2, 7, 11 | mp3an 1457 | 1 ⊢ 𝑀 ∉ Mgm |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∉ wnel 3038 ∃wrex 3062 {crab 3424 (class class class)co 7401 1c1 11107 + caddc 11109 · cmul 11111 2c2 12264 ℤcz 12555 ↾s cress 17172 Mgmcmgm 18561 ℂfldccnfld 21228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-mgm 18563 df-cnfld 21229 |
This theorem is referenced by: (None) |
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