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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnsgrpnmnd | Structured version Visualization version GIF version |
Description: The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.) |
Ref | Expression |
---|---|
nnsgrp.m | ⊢ 𝑀 = (ℂfld ↾s ℕ) |
Ref | Expression |
---|---|
nnsgrpnmnd | ⊢ 𝑀 ∉ Mnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 11835 | . . . 4 ⊢ ℕ ⊆ ℂ | |
2 | nnsgrp.m | . . . . 5 ⊢ 𝑀 = (ℂfld ↾s ℕ) | |
3 | 2 | cnfldsrngbas 44999 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
5 | nnex 11836 | . . . 4 ⊢ ℕ ∈ V | |
6 | 2 | cnfldsrngadd 45000 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
8 | 4, 7 | isnmnd 18177 | . 2 ⊢ (∀𝑧 ∈ ℕ ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd) |
9 | 1nn 11841 | . . . 4 ⊢ 1 ∈ ℕ | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℕ) |
11 | oveq2 7221 | . . . . 5 ⊢ (𝑥 = 1 → (𝑧 + 𝑥) = (𝑧 + 1)) | |
12 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
13 | 11, 12 | neeq12d 3002 | . . . 4 ⊢ (𝑥 = 1 → ((𝑧 + 𝑥) ≠ 𝑥 ↔ (𝑧 + 1) ≠ 1)) |
14 | 13 | adantl 485 | . . 3 ⊢ ((𝑧 ∈ ℕ ∧ 𝑥 = 1) → ((𝑧 + 𝑥) ≠ 𝑥 ↔ (𝑧 + 1) ≠ 1)) |
15 | nnne0 11864 | . . . . 5 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0) | |
16 | 15 | necomd 2996 | . . . 4 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧) |
17 | 1cnd 10828 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℂ) | |
18 | nncn 11838 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
19 | 17, 17, 18 | subadd2d 11208 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ (𝑧 + 1) = 1)) |
20 | 1m1e0 11902 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → (1 − 1) = 0) |
22 | 21 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ 0 = 𝑧)) |
23 | 19, 22 | bitr3d 284 | . . . . 5 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) = 1 ↔ 0 = 𝑧)) |
24 | 23 | necon3bid 2985 | . . . 4 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) ≠ 1 ↔ 0 ≠ 𝑧)) |
25 | 16, 24 | mpbird 260 | . . 3 ⊢ (𝑧 ∈ ℕ → (𝑧 + 1) ≠ 1) |
26 | 10, 14, 25 | rspcedvd 3540 | . 2 ⊢ (𝑧 ∈ ℕ → ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥) |
27 | 8, 26 | mprg 3075 | 1 ⊢ 𝑀 ∉ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∉ wnel 3046 ∃wrex 3062 Vcvv 3408 ⊆ wss 3866 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 0cc0 10729 1c1 10730 + caddc 10732 − cmin 11062 ℕcn 11830 Basecbs 16760 ↾s cress 16784 +gcplusg 16802 Mndcmnd 18173 ℂfldccnfld 20363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-addf 10808 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-mnd 18174 df-cnfld 20364 |
This theorem is referenced by: (None) |
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