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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 44920, and even a non-unital ring, see 2zrng 44911. (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 44783 | . 2 ⊢ 1 ∈ 𝑂 |
3 | 1 | 2nodd 44784 | . . 3 ⊢ 2 ∉ 𝑂 |
4 | 1p1e2 11784 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | neleq1 3058 | . . . 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 234 | . 2 ⊢ (1 + 1) ∉ 𝑂 |
8 | oddinmgm.r | . . . 4 ⊢ 𝑀 = (ℂfld ↾s 𝑂) | |
9 | 1, 8 | oddibas 44785 | . . 3 ⊢ 𝑂 = (Base‘𝑀) |
10 | 1, 8 | oddiadd 44786 | . . 3 ⊢ + = (+g‘𝑀) |
11 | 9, 10 | isnmgm 17907 | . 2 ⊢ ((1 ∈ 𝑂 ∧ 1 ∈ 𝑂 ∧ (1 + 1) ∉ 𝑂) → 𝑀 ∉ Mgm) |
12 | 2, 2, 7, 11 | mp3an 1459 | 1 ⊢ 𝑀 ∉ Mgm |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1539 ∈ wcel 2112 ∉ wnel 3053 ∃wrex 3069 {crab 3072 (class class class)co 7143 1c1 10561 + caddc 10563 · cmul 10565 2c2 11714 ℤcz 12005 ↾s cress 16527 Mgmcmgm 17901 ℂfldccnfld 20151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-addf 10639 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-fz 12925 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-starv 16623 df-tset 16627 df-ple 16628 df-ds 16630 df-unif 16631 df-mgm 17903 df-cnfld 20152 |
This theorem is referenced by: (None) |
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